Landslides - Causes, Types and Effects.pdf

288 Jiří Nedomaunique solution and that the problem is well-posed. By the well-posedness we will meanthe existence, uniqueness, and continuous dependence of the solution on the given data(T 0 ,T 1 ,W,q,u 0 , F, P). From the physical point of view, it reflects the fact that the solutiononly changes a little for little changes in the displacements and surface body forces andtemperatures.Theorem 1 Let (2), (4) hold. Let W ι ∈ L 2 (Ω ι ), q ∈ L 2 (Γ τ ), Fi ι ∈ L2 (Ω ι ), P i ∈ L 2 (Γ τ ),u 0i ∈ H 1 2 (Γ u ),T 1 ∈ L 2 ( 2 Γ u ), T 0 ∈ H 1 (Ω),gckl ∈ L 2 (Γ klc ) then there exists one weak solutionof the problem with or without friction (62)-(63). Furthermore, there exist constantsc 0 ,c 1 independent of gckl such that‖T ‖ 1 ≤ c 0 (‖q‖ L 2 (Γ τ ∪ 1 Γ u ) + ‖W ‖ L 2 (Ω) + ‖T 1 ‖ L 2 ( 2 Γ u ) + ‖T 0 ‖ 1 ),‖u‖ 1 ≤ c 1 (‖P‖ L 2,N (Γ τ ) + ‖F‖ L 2,N (Ω) + ‖u 0 ‖ 1 )The proof is similar of that of [42, 43, 48].To prove the existence and uniqueness of the variational (weak) solution for the semicoercivecase we introduce the set of all rigid displacements and rotationsP = ∪ r ι=1P ι ,P ι = {v ι |v ι = a + b × x},where a ι i ∈ R1 , b ι ∈ R 1 , i =1,...,N, ι =1,...,r, are arbitrary and the set of bilateraladmissible rigid displacementsP 0 = {v ∈ K ∩ P |v ∈ P 0 ⇒−v ∈ P 0 } == {v ∈ P V = P ∩ V |[v n ] kl =0on ∪ k,l Γ klc }.Lemma 1 Let Ω ι ⊂ R N , u ι ∈ [H 1 (Ω ι )] N , ι =1,...,r. Thene ij (u ι )=0, ∀i, j =1,...,N ⇐⇒ u ι = a ι +(b ι × x), (66)where a ι is a vector and b ι is a real constant.Lemma 2 Let Γ u ≠ ∅. ThenP V ≡ P ∩ V 0 = {0},i.e. only the zero function lies in the intersection P ∩ V 0 .For the proof see [41].Lemma 3 Since e ij (v)=0∀v ∈ P , ∀i, j, thenMoreover, if w ∈ W,e ij (w)=0∀i, j, then w ∈ P .a(u, v)=0 ∀v ∈ P. (67)Lemma 4 Let there exist a weak solution of the problem (P). ThenS(w) ≤ j gn (w) ∀w ∈ K ∩ P, i.e.∫∫∫F i w i dx + P i w i ds −ΩΓ τ∪ k,l Γ kl cg klc |[w t ] kl |ds ≤ 0 ∀w ∈ K ∩ P. (68)