Philippe Clément

Quasilinear elliptic equations with critical exponents

has no solution if Ω ⊂ R , N ≥ 3, is bounded and starshaped with respect to some point, and 2∗ = 2N/(N − 2). In (P0) the nonlinear term is a power of u with the critical exponent (N + 2)/(N − 2). This terminology comes from the fact that the continuous Sobolev imbeddings H 0 (Ω) ⊂ L(Ω), for p ≤ 2∗ and Ω bounded, are also compact except when p = 2∗. This loss of compactness reflects in that the functional whose Euler–Lagrange equation is (P0) fails to satisfy the Palais–Smale condition. Later Brezis and Nirenberg [BN] observed that the Palais–Smale condition fails at certain levels only. Then they proved that if the nonlinear term is slightly perturbed, the new problem has a solution.

Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations

Abstract. In this paper we establish a geometric theory for abstract quasilinear parabolic equations. In particular, we study existence, uniqueness, and continuous dependence of solutions. Moreover, we give conditions for global existence and establish smoothness properties of solutions. The results are based on maximal regularity estimates in continuous interpolation spaces. An important new ingredient is that we are able to show that quasilinear parabolic evolution equations generate a smooth semiflow on the trace spaces associated with maximal regularity, which are the natural phase spaces in this framework.

Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow

A time-dependent model corresponding to an Oldroyd-B viscoelastic fluid is considered, the convective terms being disregarded. Global existence in time is proved in Banach spaces provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using again an implicit function theorem.

On a class of degenerate elliptic operators arising from Fleming-Viot processes

Abstract. We are dealing with the solvability of an elliptic problem related to a class of degenerate second order operators which arise from the theory of Fleming-Viot processes in population genetics. In the one dimensional case the problem is solved in the space of continuous functions. In higher dimension we study the problem in

Some Remarks on Maximal Regularity of Parabolic Problems

L^2

Schauder estimates for a class of second order elliptic operators on a cube

spaces with respect to an explicit measure which, under suitable assumptions, can be taken invariant and symmetrizing for the operators. We prove the existence and uniqueness of weak solutions and we show that the closure of the operator in such

An elementary proof of the triangle inequality for the Wasserstein metric

L^2

Qualitative properties of solutions of volterra equations in Banach spaces

spaces generates an analytic

Global Smooth Solutions to a Fourth-order Quasilinear Fractional Evolution Equation

C_0

A bifurcation problem for point interactions in L2(ℝ3)

-semigroup.