By Michel Chipot

The target of this ebook is to introduce the reader to assorted subject matters of the speculation of elliptic partial differential equations by way of warding off technicalities and refinements. except the elemental idea of equations in divergence shape it comprises topics similar to singular perturbation difficulties, homogenization, computations, asymptotic behaviour of difficulties in cylinders, elliptic structures, nonlinear difficulties, regularity thought, Navier-Stokes process, p-Laplace equation. only a minimal on Sobolev areas has been brought, and paintings or integration at the boundary has been rigorously kept away from to maintain the reader's cognizance at the attractiveness and diversity of those matters.

The chapters are fairly self sustaining of one another and will be learn or taught individually. various effects awarded listed below are unique and feature no longer been released somewhere else. The e-book could be of curiosity to graduate scholars and school contributors focusing on partial differential equations.

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45) exists for v smooth it is enough to note that by the mean value theorem and the chain rule one has d v(x + thν) θ (θ ∈ (0, 1)) dt = ∇v(x + θhν) · hν v(x + hν) − v(x) = (recall that the · denotes here the Euclidean scalar product). 3. Sobolev Spaces 25 Dividing by h and letting h → 0 it follows that v(x + hν) − v(x) = ∇v(x) · ν = ∂ν v(x). 47) This derivative in the ν-direction will be denoted in the following as above as ∂ν v ∂v or also ∂ν . Note that if v ∈ H 1 (Ω) then for a ﬁxed vector ν, the last equality deﬁnes a function ∂ν v which is in L2 (Ω).

For any “smooth” vector ﬁeld ω in Ω we have Ω div ω dx = (dσ(x) is the measure area on ∂Ω). 20) 48 Chapter 4. Elliptic Problems in Divergence Form Note that in this theorem we do not precise what “smooth” means. 20) holds when one can make sense of the diﬀerent quantities n, dσ(x), the integrals, div ω occurring – see [11], [29], [44]. 6) with ω = ∇u. 2 in a strong form we consider only the case where V = H 1 (Ω), f ∈ L2 (Ω). Taking then v = ϕ ∈ D(Ω) we derive as above that − div(A(x)∇u) + au = f in D (Ω).

This will be a consequence of the following theorem. Before to state it let us introduce brieﬂy the notion of directional derivative. Let ν be a unit vector in Rn . 45) exists and it is called the derivative of v in the direction ν. For instance ∂x1 v is the derivative in the direction e1 = (1, . . , 0), where e1 is the ﬁrst vector of the canonical basis in Rn . 45) exists for v smooth it is enough to note that by the mean value theorem and the chain rule one has d v(x + thν) θ (θ ∈ (0, 1)) dt = ∇v(x + θhν) · hν v(x + hν) − v(x) = (recall that the · denotes here the Euclidean scalar product).