By Graves L. (ed.)
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Extra resources for Calculus of Variations and Its Applications
34) be uniformly elliptic with aij ∈ C 1 (Ω) or aij ∈ ¯ c0 ∈ L∞ (Ω). 42) (which coincide for v ∈ H01 (Ω)). 46) where C depends on Ω but not on f and y. Proof See for example  or . 41) with Robin boundary conditions. 37) satisfies in addition y H 1 (Ω) ≤C f L2 (Ω) . y H 2 (Ω ) ≤C f L2 (Ω) . 41) for Robin boundary condition satisfies y H 2 (Ω ) ≤ C( f L2 (Ω) + g L2 (∂Ω) ). If the coefficients are more regular we can iterate this argument to obtain higher interior regularity. 26 aij ∈ C m+1 (Ω), c0 ∈ C m (Ω) and f ∈ H m (Ω) hold with some m ∈ N0 .
58) must hold. 58) is positive on E. 60) would not hold for v(t) = 1E (t)w. 58). 60)) means that yt + Ly = f holds in L2 (0, T ; V ∗ ). 4 Existence and Uniqueness of Weak Solutions Let V →H →V∗ be a Gelfand triple. 17 We recall that the imbedding H → V ∗ is given by v, w (v, w)H for all v ∈ H , w ∈ V . a. 62) y(0) = y0 . 34 (i) V → H → V ∗ is a Gelfand triple, H, V separable Hilbert spaces. a. 64) 44 S. a. t ∈ (0, T ). 65) The mappings t → a(v, w; t) ∈ R are measurable for all v, w ∈ V . (iii) y0 ∈ H , f ∈ L2 (0, T ; V ∗ ).
Since AM is compact, the sequence (Axk )K possesses a convergent subsequence (Axk )K → y. The continuity of the norm implies y − Ax But since (Axk )K − contradiction. Y ≥ ε. 1 Weak Solutions of Elliptic PDEs In this section we sketch the theory of weak solutions for elliptic second order partial differential equations. , to [4, 47, 90, 115, 133, 146]. 14) where Ω ⊂ Rn is an open, bounded set and f ∈ L2 (Ω). g. source terms f that act only on a subset of Ω. Since a ¯ exists at best for continuous right hand sides, classical solution y ∈ C 2 (Ω) ∩ C 1 (Ω) we need a generalized solution concept.
Calculus of Variations and Its Applications by Graves L. (ed.)