The theory of elliptic boundary problems is fundamental in analysis and the role of spaces of weakly differentiable functions (also called Sobolev spaces) is essential in this theory as a tool for analysing the regularity of the solutions.

This book offers on the one hand a complete theory of Sobolev spaces, which are of fundamental importance for elliptic linear and non-linear differential equations, and explains on the other hand how the abstract methods of convex analysis can be combined with this theory to produce existence results for the solutions of non-linear elliptic boundary problems. The book also considers other kinds of functional spaces which are useful for treating variational problems such as the minimal surface problem.

The main purpose of the book is to provide a tool for graduate and postgraduate students interested in partial differential equations, as well as a useful reference for researchers active in the field. Prerequisites include a knowledge of classical analysis, differential calculus, Banach and Hilbert spaces, integration and the related standard functional spaces, as well as the Fourier transformation on the Schwartz space.

There are complete and detailed proofs of almost all the results announced and, in some cases, more than one proof is provided in order to highlight different features of the result. Each chapter concludes with a range of exercises of varying levels of difficulty, with hints to solutions provided for many of them.

condition for the continuity of a linear functional that uses the fundamental system of neighborhoods of 0 deﬁned using the family of seminorms {ηλ } that generate the topology of a locally convex space X. Proposition 1.64. A linear functional T on X is continuous if and only if ∃ λ, ∃ M > 0, ∀ x ∈ X, |T (x)| M ηλ (x). The importance of the existence of countable fundamental systems of neighborhoods in locally convex spaces is clear in the following two propositions. The second one

set with ﬁnite Lebesgue measure. Show that if p then we have Lp (Ω) −→ Lq (Ω). q, (2) Use a counterexample to show that this is false if Ω has inﬁnite measure. (3) Let Ω be an arbitrary open set. Show that p r q =⇒ Lp (Ω) ∩ Lq (Ω) −→ Lr (Ω). Also show that ∀ f ∈ Lp (Ω) ∩ Lq (Ω), f sup( f r p, f q ). Exercise 1.11 (The Limit of Lp Norms when p → +∞). Recall the deﬁnition of L∞ (Ω) and of the norm · ∞ on this space. Prove that if f ∈ L∞ (Ω) ∩ Lr (Ω) for at least one index r 1, then lim

gives γ0 (ϕi u) 1 + ∇ai C Lp (Oi ,dσi ) 1/p 2 ∞ ∂N (ϕi u) Lp (Ωi ) . By condition (3) of Deﬁnition 2.65, this leads to the inequalities γ0 u Lp (∂Ω) C sup i 1 + ∇ai 2 ∞ ∇(ϕi u) Lp (Ωi ) i u∇ϕi + ϕi ∇u C 1/p Lp (Ωi ) i C sup{ ϕi i ∞, ∂ N ϕi ∞} u W 1,p (Ωi ) . i Using condition (2.66), we deduce that there exists a constant C ∗ that does not depend on the elements of Deﬁnition (2.65), such that ∀ u ∈ C ∞ (Ω) ∩ W 1,p (Ω), γ0 u Lp (∂Ω) C∗ u W 1,p (Ω) . We have thus

continuous on almost all lines parallel to Ox and such that its derivative almost everywhere ∂x1 u is an element of Lp (Ω). Show that [∂x1 u] = ∂x1 u almost everywhere. (3) Let u ∈ W 1,1 (Ω). Suppose that [x, x + h] ∈ Ω. Show that the derivative of v : t → u(x + th) exists almost everywhere on ]0, 1[ and that dv/dt (x + th) = h · ∇u(x + th). Hints. For (2), it suﬃces to compute Ω ϕ∂x1 udx by integrating by parts. 106 2 Sobolev Spaces and Embedding Theorems For (3), use the decomposition of

ﬁnite. Consider the integral Ij . By setting x = xj , it becomes 1 Ij = RN 0 t∈supp ρ u(x ) − u(x − ytj+1 ej+1 ) p dt dx dy. y Substituting the variable zj+1 = ytj+1 in the partial integral with respect to y and applying Lemma 3.27, we obtain 1 Ij = 0 Cj RN t∈supp ρ Kj+1 RN −Kj+1 Cj u pj+1,1−1/p u(x ) − u(x − ytj+1 ej+1 ) p dt dx dy y u(x ) − u(x − zj+1 ej+1 ) p dzj+1 dx zj+1 < +∞. We can now conclude that all the derivatives ∂i U belong to Lp (RN × ]0, 1[). 3.5 Higher Order