A Mini-course on Elliptic–Hyperbolic Equations - ICMS

A <strong>Mini</strong>-<strong>course</strong> on Elliptic–Hyperbolic EquationsThomas H. OtwayICMS Workshop in Differential Geometry & ContinuumMechanics17–21 June 2013

Goal of the <strong>course</strong>: To illustrate the variety of contexts in whichelliptic–hyperbolic equations arise, the variety of methods used toattack them, and the variety of ideas from analysis, geometry, andphysics that contribute to understanding them.

Part I. Orientation◮ Why study this subject?◮ The analytic approach versus geometric approach◮ The linear theory versus the nonlinear theory◮ The hodograph transformation◮ The partial hodograph transformationPart II. Elliptic–hyperbolic boundary value problems◮ Open versus closed boundary conditions◮ What is the natural boundary geometry?◮ Weak solutions◮ “Almost-correct” boundary conditions

Naive discussion of equation type:Physically, what do solutions of partial differential equations do?In general, they◮ propagate as waves◮ diffuse as heat◮ do nothing at all (e.g., “oscillate”)This leads to classification of the corresponding equations as,respectively◮ hyperbolic◮ parabolic◮ elliptic

But, for example:◮ Separating variables in Laplace’s equation in toroidalcoordinates produces a partial differential equation which iselliptic inside the unit circle and hyperbolic on theR 2 -complement of the unit circle;[B. Riemann, Partielle Differentialgleichungen, 1861; also C.Neumann, 1864; W. M Hicks, 1881; E. Heine, 1881; A. B.Basset, 1893]◮ the equations for a steady, irrotational, isentropic,compressible flow are elliptic for velocities below the speed ofsound in the medium, and hyperbolic at velocities above thespeed of sound;[H. Bateman, Proc. R. Soc. London Ser. A 125 (1929)]

◮ The continuity equations for shallow water are elliptic whenthe flow speed is below a certain value (the Froude number)and hyperbolic when the speed exceeds that value;[one space dimension: D. Riabouchinsky, C. R. Acad, Sci.Paris 195 (1932); two space dimensions: J. J. Stoker,Commun. Pure Appl. Math. 1 (1948)]◮ the governing equations in a continuum model for two-lanetraffic flow in opposing directions change from hyperbolic toelliptic type over a region of phase space, provided aparameter is introduced which represents a degree ofdependence between the flow density in each lane and theflow density in the opposing lane – the area of the ellipticregion depends on the strength of the interaction parameter;[J. H. Bick and G. F. Newell, Quart. Appl. Math. 18, (1960)]

◮ the governing equations for ideal magnetohydrodynamic flowin a nozzle change between elliptic and hyperbolic type threetimes: at the sound speed c, the Alfvén speed a, and thespeed: ca/ √ c 2 + a 2 ;[C. K. Chu, Phys. Fluids 5 (1962)]◮ the complex eikonal equation is of hyperbolic type in theinterior of a circular caustic and of elliptic type on the exteriorof the caustic;[Yu. A. Kravtsov, Radiofizika 7 (1964); also D. Ludwig,Commun. Pure Appl. Math. 19 (1966); R. Magnanini and G.Talenti, Contemporary Math. 283, (1999)]

◮ the equations for a magnetic field applied to azero-temperature plasma are elliptic at certain frequencies ofthe applied field and hyperbolic at other frequencies;[for electrostatic waves: A. D. Piliya and V. I. Fedorov, Sov.Phys. JETP 33 (1971); for fully electromagnetic waves: H.Weitzner, Commun. Pure Appl. Math. 38 (1985)]◮ the governing equations of a simple model for longitudinal (orfor pure shearing) motions are strictly hyperbolic undertension and of mixed equation type under compression;[J. L. Ericksen, J. Elasticity 5 (1975)]

◮ the Laplace–Beltrami equation on the extended projective discP 2 is elliptic inside the Beltrami disc but hyperbolic outside ofit;[L-K. Hua, Proc. 1980 Beijing Symp. Diff. Geometry Diff.Equations 1 (1982); some hints in the work of G. G. Stokes,1854]◮ by a series of approximations, the partition function forquantum electrodynamics can be reduced to a relativisticmodel in which the quarks are coupled to a pair of classicalgauge fields. Additional choices reduce the analysis to aproblem in nonlinear electrostatics which change from ellipticto hyperbolic type depending on the value of a (highlynonlinear) function of the flux, its gradient, and the radialcoordinate;[S. L. Adler and T. Piran, Phys. Lett. 113B (1982)]

◮ according to the Hartle–Hawking model, the Laplace–Beltramiequation on a matter-free region of space-time would havebeen of elliptic type in the early universe and would havechanged to an equation of hyperbolic type across ahypersurface which is space-like when viewed from the present;[J. B. Hartle and S. W. Hawking, Phys. Rev. D 28 (1983)]◮ some models for multiphase flow change from hyperbolic toelliptic type;[Review: B. L. Keyfitz, Change of type in simple models oftwo-phase flow, 1992.]

◮ In certain models for plasma flow in a tokamak, the governingequation changes its type twice: from elliptic to hyperbolicand back again, with increasing flow velocity. However, thewidth of the hyperbolic region (whether negligible or not) inat least some of the models is a matter of controversy.[A. Lifschitz and J. P. Goedbloed, J. Plasma Phys. (1997);also E. Hamieri, L. Guazzotto, unpublished, 2012]◮ the wave equation in Minkowski space-time, in a referenceframe rotating with constant angular velocity with respect toanother reference frame, is elliptic for certain values of theradial coordinate and hyperbolic for other values;[J. M. Stewart, Classical Quant. Grav. 18 (2001)]

◮ the helically reduced wave equation on a torus (a qualitativelinear model for the reduction of the Einstein equations by ahelical Killing vector field) is elliptic for certain values of theradial coordinate and hyperbolic for other values;[C. G. Torre, J. Math. Phys. 44 (2003); C. Klein, Phys. Rev.D 70 (2004)]◮ An equation arising in the design of automobile windshieldschanges from elliptic to hyperbolic type as a certain shapeparameter of the windshield is varied;[N. Bîlă, SIAM J. Appl. Math. 65, (2004)]

◮ The relevant equation for the existence of an isometricembedding of Riemannian surfaces in higher-dimensionalEuclidean space is either hyperbolic or elliptic depending onthe sign of the curvature (the same is true for the equationgoverning the existence of a surface having prescribedGaussian curvature);[Review: Q.Han and J.-X.Hong, Isometric Embedding ofRiemannian Manifolds in Euclidean Spaces (2006) – theproblem goes back one way or another to Schlaefli in 1873]etc., etc., etc.... It is quite common for solutions to change theirnature (oscillatory to propagating) across a smooth curve in theirdomain. The associated partial differential equations are calledelliptic–hyperbolic.

Two approaches to elliptic–hyperbolic equations◮ basically analytic (tends to arise in continuum mechanics)◮ basically geometric (tends to arise in relativistic mechanics)Analytic approach − applies to a class of equations on a fixed,generic domain.Consider a fixed domain Ω ⊂ R n (in this <strong>course</strong> we almost alwaystake n = 2) and the class of differential operators having thefollowing higher-order terms:Lu = αu xx + 2βu xy + γu yyα, β, and γ are given functions of the coordinates (x, y) ∈ Ω; u isan unknown function of x and y.If ∆ ≡ αγ − β 2 < 0, then the characteristic linesαdy 2 − 2βdxdy + γdx 2 = 0are real-valued and solutions propagate as waves.

Adopting the “differential polynomial” approach to classification,we replace the second-order derivatives by quadratic algebraicterms and obtain a classification for differential operatorsanalogous to the classification of quadratic surfaces:∆ = αγ − β 2 ⎧⎨⎩corresponding to “elliptic,” “hyperbolic,” “parabolic” equations,respectively.If ∆ changes sign on a smooth “sonic” (“parabolic”) curve on thedomain, then the equation associated with L is said to be of mixedelliptic–hyperbolic type.>

A linear example [Lavrent’ev & Bitsadze, Dokl. Akad. Nauk 20(1950)]:Lu = sgn[y]u xx + u yy = 0,for which the sonic curve is the x-axis.Laplace’s equation on the upper half-planeWave equation on the lower half plane.A quasilinear example [S-X. Chen, Acta Math. Sci. 31B (2011)]:sgn[u]u xx + u yy = 0,a model for the reflection of certain shock waves in compressiblegas dynamics.In the nonlinear case the sonic curve cannot be identified withoutsolving the equation.

Example of a first-order elliptic–hyperbolic systemA scalar equation arising in, e.g., nonlinear elasticitywhere x ∈ R and t ∈ R + .u tt = σ (u x ) x, (1)Assumptions on the stress function σ that lead to a change inequation type: ∃ α, β, γ, and δ such that, e.g.,

This is equivalent to the systemw t = v x , v t = σ(w) x .We can impose initial values w (x, 0) = w 0 (x) and v (x, 0) = v 0 (x).Equating mixed partial derivatives of v yieldsw tt = σ(w) xx ,and making the substitutions w = u x and v = u t , yieldsu tt = σ (u x ) x,which is eq. (1) of the preceding frame.[Pego and Serre, SIAM J. Numer. Anal. 25 (1988); Boonkasameand Milewski, Studies in Appl. Math. 128 (2011); Bialy andMironov, arXiv e-print, December 2012]

We obtain from the first-order system the matrix equation( ) ( ) ( )w 0 1 w=v σ ′ .(w) 0 vtThis is of <strong>course</strong> a matrix equation of the form X t = AX x . Nowadopting the “eigenvalue approach” to classification, we computethe eigenvalues of the matrix −A by solving the scalar equation|A + λI | =∣Because the eigenvaluesλ 1σ ′ (w) λλ = ± √ σ ′ (w)∣ = λ2 − σ ′ (w) = 0.are real-valued and nondegenerate when σ ′ is positive, in that casethe characteristics are real and we take this to be the region ofhyperbolicity for the system.Remark: classical solutions of this equation tend to havefinite-time singularities even in the strictly hyperbolic regime, so weusually look for weak solutions.x

Making the choice σ(t) = t 2 in the preceding discussion leads toequations of Boussinesq type, which arise in hydrodynamics. Weview these as first-order systems u t + B(u)u x = 0, takingu = (h, w) T , whereB(u) =(−hw1−w 21−h222−hwThe sub-domain of hyperbolic solutions, that is, the domain ofwave motion, coincides with that region of the domain on whichthe eigenvalues of B, that is, the numbers)√(1 − hλ ± B = −hw ± 2 ) (1 − w 2 ),2are real and the associated eigenvectors span R 2 . That correspondsto the region in which |h| and |w| are both less than 1. This is asquare, not in the xt-plane but in the “phase space” of thehw-plane – a square which cannot be determined without solvingthe system..

Geometric approach: replace the class of differential operators Lon a fixed, generic domain with a fixed, generic differentialoperator L on a class of domains (M, g) .Laplace–Beltrami operator:L g u = 1 √|g|∂∂x i (g ij√ |g| ∂u∂x j ),on a class of manifolds for which the matrix g ij represents the localmetric tensor having determinant g.

Geometric interpretation of the (linear) Lavrent’ev–BitsadzeequationLu = sgn[y]u xx + u yy = 0,g is Euclidean above the x-axis:with signature (+, +)ds 2 = dx 2 + dy 2 ,and Minkowskian below the x-axis:with signature (+, −) .In general:ds 2 = dx 2 − dy 2 ,◮ elliptic operators ⇐⇒ Riemannian (“Euclidean”) metrics◮ hyperbolic operators ⇐⇒ semi-Riemannian (typically,Lorentzian) metrics.

Four features of the geometric approach:◮ The type of a linear second-order equation is not a function ofthe associated linear operator at all, as that operator is alwaysthe Laplace–Beltrami operator. Instead, the type of theequation is a feature of the metric tensor on the underlyingpseudo-Riemannian manifold.◮ Any change in signature which results in a change in sign ofthe metric determinant g will change the Laplace–Beltramioperator on the metric from elliptic to hyperbolic type.◮ In order to decide which boundary value problems are naturalfor a second-order linear elliptic–hyperbolic equation, oneshould study the geometry of the underlying metric.◮ Any curve on which the change of type occurs will necessarilyrepresent a singularity of the metric tensor, as g will vanishalong that curve. However, none of the terms in the equationitself need blow up on this curve. (An example is the waveequation on extended P 2 .)

The linear theoryWhy bother with the linear theory of elliptic–hyperbolic equations?Two arguments against emphasizing on the linear theory:“The linear theory of elliptic equations extends in a qualitativesense to much of the nonlinear theory, but nonlinear hyperbolicequations are much more singular than in the linear case, so thelinear elliptic–hyperbolic theory is not much of a guide to nonlinearequations, which are the ones of real interest.”“In the important applications, linear elliptic–hyperbolic equationsare only associated with small initial data.”Five reasons for developing a sound linear theory:

1. Elliptic–hyperbolic continuity equations tend to arise in specialforms which allow linearization by the hodograph method. Theseequations are more ubiquitous than people think (see below), andthe hodograph method is more useful than people think (see, forexample, the work of Yuxi Zheng)2. In, e.g., plasma contexts, linear elliptic–hyperbolic models arisepartly from certain physical hypotheses, for example:◮ magnetically dominated low-beta fields (“force-free models”)which arise in models of the solar corona, ball lightning, andintergalactic jets;◮ fields in which small-amplitude electromagnetic wavespropagate with a velocity much greater than the thermaloscillations of the particles (“cold plasma models”), whicharise in certain models of tokamak plasmas;◮ equilibrium models, in which a steady plasma is surrounded bya vacuum [R. Temam, Arch. Rat. Mech. Anal. 60, (1975)].

3. The presence of an elliptic region can exert a regularizing effecton the hyperbolic region (e.g., elliptic–hyperbolic models ofcaustics in which boundary data can be prescribed in the ellipticregion, which surrounds the hyperbolic region), making theanalogy to nonlinear hyperbolic pde less apt.4. Few experts in nonlinear waves would prefer not to have a goodlinear theory of hyperbolic equations.5. An understanding of the linear theory is necessary in order tosolve nonlinear partial differential equations locally byimplicit-function methods:

Denote by Ω a domain of R n and consider the solvability of anonlinear partial differential equation F (u) = f , where f lies in thefunction space H k (Ω). If D p is a fixed open set of H p , then wewant to solve this equation for u ∈ D p . It is generally possible tofind an approximate solution u 0 such that F (u 0 ) is “close” to f insome useful sense. The goal is to use some version of the ImplicitFunction Theorem to perturb u 0 locally into a local solution to theequation F (u) = f . The solvability of the nonlinear equationdepends critically on the invertibility at u 0 of the linear operatorF ′ (u) v = d dt |t=0F (u + tv) .(There are technical problems connected with this method.) Twoimportant classes of nonlinear elliptic–hyperbolic problems ingeometry have been attacked in this way.

The isometric embedding problem: When can a 2-dimensionalRiemannian manifold be “seen” via a local realization in R 3 ? Thisproblem is associated with the existence of a local solution to thefully nonlinear Darboux equation. But that equation has beensuccessfully approached via Nash–Moser methods involving thelinearization(aKu x ) x+ bu yy + cKu x + du y ,where a, b > 0 and K is the Gaussian curvature of the surface.[Han, Commun. Pure Appl. Math. 58 (2005); Han and Khuri,Commun. Anal. Geom. 18 (2010)]

Given a function K (x, y) defined in a neighborhood of the origin,does there exist a graph z (x, y) having Gaussian curvature K?Because every surface may be expressed locally as a graph, this isthe problem of locally prescribing the curvature of a surface. As inthe preceding geometric problem, this problem can be reduced tothe local solvability of a Monge–Ampère equation, and in this caseas well implicit-function methods lead one to consider the linearequationu yy + ( x 2 − y 2) u xx = f .[Q. Han and M. Khuri, Calc. Var. Partial Diff. Eqs. (2012)]

Local canonical forms for linear equationsThe two canonical forms for linear, second-order elliptic–hyperbolicequations were discovered by Francesco Tricomi (left) and MariaCinquini-Cibrario (right).

For most of the past century, the study of linear elliptic–hyperbolicequations was generally identified with the study of the Tricomiequationyu xx + u yy = 0.Typical domain: Ω ⊂⊂ R 2 bounded by a smooth Jordan curve γin the upper half plane and the characteristic linesΓ 1 : x − 2 3 (−y)3/2 = −1, x < 0,in the lower half-plane.Γ 2 : x + 2 3 (−y)3/2 = 1, x > 0,

Tricomi [Rend. Accad. Naz. Lincei Ser. 5 14 (1923)]:◮ A unique solution exists on Ω provided the values of u (x, y)are smoothly prescribed on the elliptic arc γ and thecharacteristic line Γ 2 (True).◮ Any sufficiently smooth equation having the formα (x, y) u xx + 2β (x, y) u xy +γ (x, y) u yy + lower order terms = 0,for which the discriminant β 2 − αγ changes sign along asmooth curve, is locally equivalent to the equationyu xx + u yy + lower order terms = 0 (False).

Tricomi’s assistant Cinquini-Cibrario realized that there wereelliptic–hyperbolic equations which are not locally equivalent toTricomi’s equation. Example:xu xx + u yy = 0.(Old “NYU” name for this equation: anti-Tricomi.)Boundary value problems have solutions on a domain which lookslike a Tricomi domain rotated clockwise by 90 ◦ . Red and bluecurves: Characteristics y = ∓ √ −x ± 4. Cinquini-Cibrario showedthat there exist classical solutions with data prescribed on theelliptic boundary.

Cinquini-Cibrario’s local canonical forms [Rend. R. Inst. Lomb.65 (1932)]:L T ≡ y 2m+1 u xx + u yy + lower-order termsorL K ≡ u xx + y 2m+1 u yy + lower-order terms.(Here m is a non-negative integer.)A differential equation for which the differential operator L can betransformed locally into an operator of the form L T is said to be ofTricomi type.A differential equation for which the differential operator L can betransformed locally into an operator of the form L K is said to be ofKeldysh type.

M.V. Keldysh had nothing to do with the discovery of either of thecanonical forms. Twenty years later he studied the degeneration ofellipticity in the class of equations introduced and studied byCinquini-Cibrario.[M. V. Keldysh, On certain classes of elliptic equations withsingularity on the boundary of the domain, Dokl. Akad. NaukSSSR 77 (1951)]

Alternative definition of the canonical forms:We can always write a second-order linear equation with smoothcoefficients defined in an open set of R 2 in the local formLu = K (x, y) u xx + u yy + lower-order terms.One can, by a further nonsingular transformation, arrive at acoordinate system in which the type-change function K (x, y) haseither of two forms:1. K(y), with K(0) = 0 and yK(y) > 0 for y ≠ 0, or2. K(x), with K(0) = 0 and xK(x) > 0 for x ≠ 0.The former characterizes equations of Tricomi type and the latter,equations of Keldysh type.Still more generally, one notices that the distinction is onlymeaningful in a neighborhood of the sonic curve.

Equations of Keldysh type can be characterized by thedegeneration of their characteristic lines, which intersect soniccurve tangentially (leading to weaker regularity):

Example of the weaker regularity of equations of Keldyshtype:Fundamental solutions are the main ingredient for solving theDirichlet problem by the Green’s function method.A fundamental solution exists for the Tricomi equation[Barros-Neto & Gelfand, Duke Math. J. 98(1999)].A fundamental solution exists for the Keldysh equation only as thefinite part of a divergent improper integral [S-X. Chen, Sci. inChina, Ser. A: Math. 52 (2009)].

Why do equations of Keldysh type have weaker regularity?Consider a differential operator of order m > 0 on an open setΩ ⊂ R n ,Pu(x) = ∑ ( ) ∂ αc α (x) u(x),∂x|α|≤mξ α = ξ α 11 · · · ξαn n , having C ∞ real (for simplicity) coefficients. Theprincipal symbol of P is the functionσ P (x, ξ) = ∑c α (x)ξ α .|α|=mIf P is of real principal type then, roughly speaking, all itssignificant properties are determined by its principal symbol.

In general, the operator P is of real principal type if at every pointx ∈ Ω its real characteristic coneC (P, x) = {ξ ∈ R n | ξ ≠ 0, σ P (x, ξ) = 0}has no singular points. (If P is elliptic, then C is empty.)There are many equivalent conditions for an operator to be of realprincipal type. In the concrete case of an operator having the formLu = K (x, y) u xx + u yy + lower-order ,the operator L is of real principal type if whenever K (x, y) = 0, wealso have K y (x, y) ≠ 0.

Let ξ = K (x, y) and η = λ (x, y) . This produces an operator ofthe formLu ≡ Ku ξξ + γu ηηwhere λ has been chosen so that the following identities aresatisfied:KK x λ x + K y λ y = 0; (2)Kλ 2 x + λ 2 y = γ. (3)If K and K y never vanish simultaneously at the sonic transitionK = ξ = 0, then λ y is forced by (2) to vanish whenever Kvanishes; that is, λ y vanishes at the sonic transition. But in thatcase, condition (3) forces γ to also vanish at the sonic transition.

Writing the characteristic equation for Lu = 0, we haveKdη 2 + γdξ 2 = 0. (4)If K and K y can vanish simultaneously, then it is possible to haveγ ≠ 0 on the sonic curve, in which case eq. (4) assumes the formγdξ 2 = 0, γ ≠ 0. So in the case K = 0 = K y the characteristiclines satisfy ξ = dξ = 0. That is, they degenerate on the soniccurve, and the case K = 0 = K y corresponds to equations ofKeldysh type.A tedious algebraic calculation will show directly that if K and K ydo not vanish simultaneously, then the operator L is of Tricomitype.That is to say, the Tricomi operator (K (x, y) = y) is of realprincipal type but the Keldysh operator (K (x, y) = x) is not, anda lower-order perturbation of a well-behaved operator of Keldyshtype can be badly behaved.

Moreover the Implicit Function Theorem requires that when thelinear operator P is inverted, all the p derivatives lost in theoriginal linearization map are regained. That condition is onlystrictly satisfied by so-called hypoelliptic operators. However, if Pis of real principal type, then an inverse exists that loses only onederivative. That case is amenable to treatment by the Nash–Moseriteration. So the “right” class of operators to consider when usingimplicit-function methods – not as restrictive as hypoellipticity, butrestrictive enough to satisfy the requirements of the Nash–Moseriteration – is the class of operators of real principal type. That is,Nash–Moser arguments are well-adapted to linearizations which areoperators of Tricomi type, because they are of real principal type,but are not well-adapted to operators of Keldysh type.

The hodograph linearizationMost of the analytic techniques for elliptic–hyperbolic equationsrequire linearity. Most of the elliptic–hyperbolic models in physicsare nonlinear. But many of these models are in the very specialform required for linearization by the hodograph method.In its full generality, the hodograph transformation is ann-dimensional method in which a function u(x), x ∈ R n , havingnonzero Hessian, is transformed locally by a mapping of the regioninto a new region given by the coordinate transformationx → y = ∇u. The derivatives of u can then be expressed in termsof the derivatives of y via the Legendre transformation u → v,wherev = x i y i − u = x i u x i − u.(Here and below, repeated indices are summed from 1 to n.) Thuspartial differential equations for u become equations for v.

If a system of equations has the form[ ] ( ) [ ] (a11 a 12 p b11 b+12 pa 22 qb 22 qa 21xb 21)y( 0=0), (5)where the entries of the coefficient matrices depend only on p andq, then the coordinate transformation (x, y) → (p, q) takes eq. (5)into a linear equation having the form[ ] ( ) [ ] ( ) ( )b12 −a 12 x −b11 a+11 x 0= .−a 22 ya 21 y 0b 22p−b 21This transformation is called a hodograph map. It fails at anypoint for which its Jacobian J = p x q y − p y q x vanishes.q

This map is easy to understand as an inversion of the differentialformsdp = p x dx + p y dyanddq = q x dx + q y dy.Written in matrix form, this is the system( ) ( ) ( dp px p=y dxdq q x q y dy),which has the inverse( ) dx= 1 dy Jprovided J is nonvanishing.−q x p x dq(qy −p y) ( dp)

The coordinate systems (p, q) and (x, y) are related by theLengendre transformationV (p, q) = xp + yq − v(x, y), (6)where in this caseand(x, y) =(p, q) =( ∂V∂p , ∂V )∂q( ∂v∂x , ∂v ).∂y

Example of the use of the hodograph method: [Magnanini &Talenti, in: Ill-posed and Inverse Problems, VSP, (2002)]( )( )|∇θ| 4 − θy2 θ xx + 2θ x θ y θ xy + |∇θ| 4 − θx2 θ yy = 0. (7)This quasilinear elliptic–hyperbolic equation arises in the context ofconstructing uniform asymptotic expansions of the Helmholtzequation near a smooth, convex caustic.We can write this equation as a homogeneous first-order system inp (x, y) and q (x, y) , with quasilinear term depending only on pand q.

Making the substitutions p = θ x , q = θ y in the original quasilinearsystem (7), we apply the hodograph transformation and eventuallyobtain the (now linear) scalar equation[ (p 2 + q 2) 2− p2 ] V pp− 2pqV pq +[ (p 2 + q 2) 2− q2 ] V qq = 0, (8)provided there is a continuously differentiable scalar functionV (x, y) for which V p = x and V q = y.

Three limitations of the hodograph linearization◮ It requires that the equation be of a very special form and isnot robust to small changes in that form.◮ The hodograph transformation may be singular. (Butgenerally, any singular set on the elliptic region of theequations must consist of isolated points.)◮ Whereas the hodograph mapping replaces a quasilinearequation by a linear one, it tends to replace linear boundaryconditions by nonlinear ones. In general this restricts theboundary conditions that can be applied in this method torelatively simple examples.Example: The vanishing of a solution to the Magnanini–Talentiequation (8) on a circle in the hodograph plane pulls back, under ahodograph transformation, to a constant value of the solution onthe corresponding circle in the physical plane.

The partial hodograph transformationThis is a(nother) very old method that has attracted new interestrecently. Instead of switching all the dependent variables in theequation with all the independent variables as in the classicalhodograph method, In this variant we switch just one. (In theory,switching any number of dependent variables that is fewer than allof them will result in a partial hodograph transformation.)This is not a linearization method. The original idea, due toFriedrichs [Math. Ann. 109 (1934)], was to use this method totransform a hypersurface on which a solution vanishes into a flatsurface in new coordinates. More generally, this idea can be usedto change a free boundary problem into one with a fixed boundary.That use of the method was apparently introduced, in the contextof elliptic theory, by Kinderlehrer and Nirenberg [Annali Sc. Norm.Sup. Pisa 4 (1977)].

Free boundary problems:It is sometimes the case that a boundary condition is not givenexplicitly but must be determined as part of the solution to theproblem. Examples of such free boundary conditions include:◮ the elevation of the ocean surface in certain hydrodynamicmodels;◮ the boundary of a plasma, held in equilibrium by an externallyapplied magnetic field, and the surrounding vacuum;◮ the upper boundary, contiguous to the ceiling, of a drop ofwater suspended from the ceiling, in which the angle betweenthe lower surface of the drop and the ceiling is given.

Example for an elliptic–hyperbolic pde [S-X. Chen and D. Li[Methods & Applic. Anal. 8(2001)]Show ∃ u satisfying the nonlinear Lavrent’ev–Bitsadze equationu xx + sgn[u]u yy = 0 on the domain Ω bounded by the smoothcurve y = γ 0 (x) on the upper half plane and the lines L 1 and L 2given by the equations y = −x and y = x − 1, respectively. ∃ βand φ for which:u = β(x) for (x, y) ∈ γ 0 , 0 ≤ x ≤ 1, (9)u = φ(x) for (x, y) ∈ L 1 , 0 ≤ x ≤ 1/2. (10)

Idea of the method: Solve the boundary value problem in thehyperbolic region Ω − for a solution u − , and then paste thissolution onto a solution u + of the problem restricted to the ellipticregion Ω + , where Ω − denotes the region on which the nonlinearLavrent’ev–Bitsadze equation is hyperbolic, and Ω + denotes theregion on which the equation is elliptic. In doing so, we requirethat the solution u − ∪ u + be continuous across the sonic curve Γ 1 .But Γ 1 cannot be found explicitly without solving the boundaryvalue problem, and thus will eventually become our free boundary.Because the existence of a C 1 curve Γ 1 such that u > 0 above Γ 1and u < 0 below Γ 1 is guaranteed by the Implicit FunctionTheorem only near u = y, the method shows the existence of aunique C 1+α solution only near u = y, where α depends on thedata.

Because the nonlinear Lavrent’ev–Bitsadze equation reduces to thewave equation below Γ 1 , on this sub-domain Ω − the equationpossesses the D’Alembert solutionu (x, y) = F (x − y) + G (x + y) . (11)Recall the boundary condition (10):u = φ(x) for (x, y) ∈ L 1 , 0 ≤ x ≤ 1/2.On the line L 1 we have, by substituting y = −x into (11) andusing the boundary condition,φ(x) = F (2x) + G(0). (12)Solving for F in (11) and substituting the result into (12) yields( ) x − yu (x, y) = φ + G (x + y) − G(0),2and thus, by direct calculation:

( ) x − yu x − u y = φ ′ . (13)2We satisfy the condition that u − transforms continuously into u +across the sonic curve by declaring that condition (13) also holdson the sonic curve Γ 1 . Now it remains only to solve the boundaryvalue problem on the subdomain Ω + . But that Tricomi problemhas become the free boundary problemu xx + u yy = 0 in Ω + ,u = β(x) > 0 on γ 0 ,u = 0 and u x − u y = φ ′ ( x − y2)on Γ 1 ,where the unknown sonic curve Γ 1 of the elliptic-hyperbolicboundary value problem has become the free boundary of anelliptic boundary value problem.

In order to fix the boundary, we now apply the partial hodographtransformation (x, y) → (ξ, z) , where ξ = x, z = u (x, y) . Interms of the inverse transformation x = ξ, y = h (ξ, z) , theoriginal equation assumes the formh 2 zh ξξ + ( 1 + h 2 ξ)hzz − 2h ξ h z h ξz = 0,with the fixed boundary conditionsh = γ 0 (ξ) on z = β(ξ), 0 ≤ ξ ≤ 1,( ) ξ − hh ξ + φ ′ h z = −1 on z = 0.2This is now an oblique derivative problem on the image of Ω +under the partial hodograph transformation. The new domain isbounded in the half-plane z > 0 by the line z = 0 and the curvez = β(ξ). The transformed boundary value problem is stillnonlinear; but it can now be attacked by the methods of elliptictheory.

Part II: What are the right boundary conditions forelliptic–hyperbolic equations?To begin with, what is the right number of boundary conditions?Example 1: Find an L 2 solution to the ODEx dudx + u = 0,u = u(x), on the interval −1 ≤ x ≤ 1. Writingd(xu) = 0,dxwe find that xu = c, or u(x) = c/x, where c is the constant ofintegration.The solution is in L 2 [−1, 1] only if c 2 /x 2 is integrable, and thatwill be the case only if c = 0; so this constant of integration canbe evaluated without placing any conditions at all at either of theboundary points x = ±1.

Example 2: Find an L 2 solution to the ODEon [−1, 1] . The solution−x dudx + 1 = 0u(x) = log |x| + c,has a singularity at x = 0. But it is in L 2 [−1, 1] :∫ 1limɛ↓0 ɛlog 2 (x)dx + limɛ↑0∫ −ɛ−1log 2 |x|dx < ∞.Because u(0) is undefined, there are actually two solutions: u + (x),which is log(x) on (0, 1] and identically zero on [−1, 0], and u − (x)which is log(−x) on [−1, 0) and identically zero on [0, 1].Conditions can thus be prescribed at both x = −1 and x = 1.

Friedrichs’ interpretation of examples like these:A boundary condition is lost in the first equation because weimpose extra smoothness.An extra boundary condition arises in the second equation becausewe allow a singularity in the domain.For elliptic–hyperbolic boundary value problems:1. Requiring the solution to extend smoothly across the sonic curvemay allow us to remove conditions on some of the boundary arcs.2. If we place conditions on the entire boundary we may stillobtain a well-posed problem if we permit the solution to besufficiently singular.

Open boundary conditions: Data are prescribed on a proper subsetof the boundary.Examples:1. Tricomi problem: solutions are prescribed on the elliptic part ofthe boundary and on a characteristic arc.2. Guderley–Morawetz problem: Solutions are prescribed on theelliptic boundary and the non-characteristic hyperbolic boundary.

Closed boundary conditions: Data are prescribed on the entireboundary.Examples: The classical Dirichlet, Neumann, and mixedDirichlet–Neumann problems. (Open versions of these mayprescribe the data on some given proper subset of the boundary.)Problem: The “ice-cream-cone-shaped” domains of Tricomi andCinquini-Cibrario are natural for open Dirichlet problems but notfor closed Dirichlet problems.Open Dirichlet problems arise naturally in gas dynamics (nozzles)but may not be natural in other applications.Closed Dirichlet problems also arise naturally in gas dynamics (flowaround an airfoil) as well as in applications to plasma dynamicsand the theory of caustics.

Geometries of canonical domains:Geometry for boundary value problems associated with the fourmain types of partial differential equations in two space dimensions.

D-star-shaped domains [Lupo & Payne, Comm. Pure Appl.Math. 56 (2003)]If a domain is star-shaped with respect to a vector field D, then itis possible to “float” from any point of the domain to the originalong the flow lines of the vector field.

Ingredients:1. A one-parameter family ψ λ (x, y) of inhomogeneous dilations()ψ λ (x, y) = λ −α x, λ −β yfor α, β, λ ∈ R + ,2. an associated family of operatorsΨ λ u = u ◦ ψ λ ≡ u λ .3. a vector field D such that[ ] dDu =dλ u λ|λ=1= −αx∂ x − βy∂ y .

An open set Ω ⊆ R 2 is said to be star-shaped with respect to theflow of D if ∀ (x 0 , y 0 ) ∈ Ω and each t ∈ [0, ∞] we haveF t (x 0 , y 0 ) ⊂ Ω, where(F t (x 0 , y 0 ) = (x(t), y(t)) = x 0 e −αt , y 0 e −βt) .If these flow lines are straight lines through the origin (α = β) ,then we recover the conventional notion of a star-shaped domain.

Moreover, whenever a domain is star-shaped with respect to theflow of a vector field satisfyingDu = −αx∂ x − βy∂ y ,the domain boundary will be starlike in the sense that(αx, βy) · ˆn (x, y) ≥ 0,where ˆn is the outward-pointing normal vector on ∂Ω. Given avector field V = − (b, c) and a boundary arc Γ which is starlikewith respect to V , the inequalityis satisfied on Γ.bn 1 + cn 2 ≥ 0

It frequently happens that weak solutions to elliptic–hyperbolicboundary value problems may exist only in weighted spaces:and(∫ ∫1/2||u|| L 2 (Ω;K) = |K|u dxdy) 2 ;Ω[∫ ∫]||u|| H 10 (Ω;K) = (|K|u2x + uy2 ) 1/2dxdy ;||u|| H −1 (Ω;K) =Ω|〈u, ξ〉|sup,0≠ξ∈C0 ∞(Ω) ||ξ|| H 10 (Ω;K)where 〈 , 〉 is the duality bracket of distribution theory.These weights vanish on the curve K = 0 along which the equationchanges type, and on which even weak solutions may be singular.

Example: the cold plasma model(x − y2 ) u xx + u yy = 0.Characteristic lines satisfy(x − y2 ) dy 2 + dx 2 = 0.

Following Lupo, Morawetz, and Payne, we define a weak solutionof an elliptic–hyperbolic equationLu ≡ (K (x, y) u x ) x+ u yy = f ,on a Lipschitz domain Ω, with boundary conditionu(x, y) = 0∀ (x, y) ∈ ∂Ω,to be a function u ∈ H0 1(Ω; K) such that ∀ξ ∈ H1 0 (Ω; K) we have∫ ∫〈Lu, ξ〉 ≡ − (Ku x ξ x + u y ξ y ) dxdy = 〈f , ξ〉,Ωwhere 〈 , 〉 is the duality pairing between H0 1 (Ω, K) andH −1 (Ω, K) . In this case the existence of a weak solution isequivalent to the existence of a sequence u n ∈ C0∞ (Ω) such that‖u n − u‖ H 10 (Ω;K) → 0 and ‖Lu n − f ‖ H −1 (Ω;K) → 0as n tends to infinity.

TheoremLet Ω be star-shaped with respect to the flow of the vector fieldV = − (mx, µy) , where µ is a positive constant and m exceeds3µ. Suppose that x is nonnegative on Ω and that the origin ofcoordinates lies on ∂Ω. Then for every f ∈ L 2 ( Ω; |K| −1) there is aunique weak solution u ∈ H0 1 (Ω; K) to the Dirichlet problemwhere K = x − y 2 .[K (x, y) u x ] x+ u yy = f (x, y) ∀ (x, y) ∈ Ω, (14)[SIAM J. Math. Anal. 42(2010)]u (x, y) = 0 ∀ (x, y) ∈ ∂ΩIf K = y, then this result is due to Lupo–Morawetz–Payne [CPAM60 (2007)] and in that case the solution exists in H 1 (Ω).

Physical interpretation: The Dirichlet problem is weakly well-posedfor the application of Maxwell’s equations to a zero-temperatureplanar plasma to which a longitudinal magnetic field has beenapplied. The mathematical model suggests a singularity at pointsat which a flux line is tangent to a resonance frequency of thefield. This singularity supports the physical conjecture of a heatingzone at such points. The technical hypotheses of the theorem aresuggested by the physical model.

Idea of the proof: Integral variant of the abc method.Original method due to Friedrichs, first published by Protter in1953. The integral variant is due to Didenko, based on theapproach of Berezanskii and the “Soviet School” of linear pde.The method was greatly extended by Lupo and Payne in a series ofpapers, eventually in collaboration with Morawetz.Suppose that we want to establish the existence of a generalizedsolution to the boundary value problemLu = f in Ω ⊂⊂ R 2 ;u = 0 on ∂Ω,where u is an unknown function of (x, y) , f is a known function of(x, y) , and L is a linear, second-order differential operator.

The method is based on establishing a fundamental inequality forsolutions u of Lu = f having the general form‖u‖ U ≤ C‖L ∗ u‖ Vfor a suitable choice of function spaces U and V , and allu ∈ C0∞(Ω).In the conventional abc-method, we then seek functions a, b, andc such that, for some δ > 0,∫ ∫(au + bu x + cu y ) L ∗ u dxdy ≥Ω∫ ∫δ |K|ux 2 + uy 2 dxdy.Ω

If we find such a, b, and c, then we reason that∫ ∫∫ ∫δ |K|ux 2 + uy 2 dxdy ≤ (au + bu x + cu y ) L ∗ u dxdyΩ≤ ‖au + bu x + cu y ‖ L 2 (Ω)‖L ∗ u‖ L 2 (Ω)≤ C ′ K‖u‖ H 10 (Ω;K) ‖L∗ u‖ L 2 (Ω),where C ′ K depends on sup (|K/K′ |) . Thus we have‖u‖ H 10 (Ω;K) ≤ C ′ K‖L ∗ u‖ L 2 (Ω),and are forced to choose U = H 1 0 (Ω; K) and V = L2 (Ω) in thefundamental inequalityΩ‖u‖ U ≤ C‖L ∗ u‖ V(which will force our eventual solution to be only in L 2 ).

Now we defineJ f (L ∗ ξ) ≡ 〈f , ξ〉 ≤ ‖f ‖ H −1 (Ω;K)‖ξ‖ H 10 (Ω;K)≤ C‖f ‖ H −1 (Ω;K)‖L ∗ ξ‖ L 2 (Ω). (15)We find that if f ∈ H −1 (Ω; K) , then J f is a bounded functional onthe subspace of L 2 consisting of elements ξ ∈ C0∞(Ω) for which L∗ ξis bounded in L 2 . Hahn–Banach arguments extend inequality (15)to the L 2 -closure and we apply the Riesz Representation Theoremin the inner product space L 2 . This allows us to show the existenceof a distribution solution in L 2 , that is, a square-integrable functionu such that ∀ξ ∈ H0 1 (Ω; K) for which L∗ ξ ∈ L 2 (Ω) ,(u, L ∗ ξ) = 〈f , ξ〉 ,where ( , ) is the L 2 inner product and 〈 , 〉 is the Lax dualitybracket.

Suppose instead that we prove the fundamental inequality usingthe integral variant of the abc method, under the hypothesis thatΩ is star-shaped with respect to the vector field − (b, c) . It can beshown that∫ ∫δΩ(|K|v2x + vy2 )dxdy ≤(v, L ∗ u) ≤ ‖v‖ H 10 (Ω;K) ‖L∗ u‖ H −1 (Ω;K) .Applying the integral abc-method as before, we eventually find thatJ f (L ∗ ξ) ≤ C||f || L 2 (Ω;|K| −1 )||L ∗ ξ|| H −1 (Ω;K).Now applying Hahn–Banach arguments as in the preceding case,we find that we are led by duality to apply the RieszRepresentation Theorem in the inner product space H0 1 (Ω; K)rather than the inner product space L 2 . This leads to the existenceof a unique, weak solution.

That is, we find that there is a u ∈ H0 1 (Ω; K) such that〈u, L ∗ ξ〉 = (f , ξ) L 2 (Ω)(16)∀ξ ∈ H 1 0 (Ω; K) where L∗ : H 1 0 (Ω; K) → H−1 (Ω; K) is a unique,continuous extension of the original operator. Note that the spaceweighted-H −1 for L ∗ ξ is appropriate for obtaining u inweighted-H 1 , as indicated by the duality bracket on the left-handside of (16). However, due to the restrictive way in which a weaksolution has been defined, we cannot actually apply this method inan obvious way unless L = L ∗ .

Instead of the L 2 solution which resulted from applying the RieszRepresentation Theorem to the abc-method in the previous case,in the present case the Riesz Representation Theorem has given usa solution in weighted-H 1 . For that reason, and because we canderive derive uniqueness, it is appropriate to call the solution weak.So the advantage of the integral variant of the abc method overthe conventional abc-method is that the former involves estimatesthat are one derivative higher than those of the latter method,leading to the application of the Riesz Representation theorem in ahigher (although weighted) Sobolev space. Because when applyingthe Riesz Representation Theorem, L ∗ ξ is dual to the solution u inthe Lax duality bracket, L ∗ ξ ∈ L 2 (Ω) implies u ∈ L 2 (Ω), whereasL ∗ ξ ∈ H −1 (Ω; K) implies u ∈ H0 1 (Ω; K) .