Olivier Guibé

Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is (formula maths) where Ω is a bounded open subset of R N , N > 2, Ap is the so-called p-Laplace operator, 1 < p < N, μ is a Radon measure with bounded variation on Ω, 0 ≤ γ ≤ p-1,0 ≤ λ ≤ p-1,and |c| and b belong to the Lorentz spaces L Ν p-1,r (Ω), N p-1 ≤r +∞, and L N,1 (Ω), respectively. In particular we prove the existence under the assumptions that γ=λ=p-1,|c| belongs to the Lorentz space LN p-1,r(Ω) N p-1≤r<+∞, and ||c|| N r is small enough.

Remarks on the uniqueness of comparable renormalized solutions of elliptic equations with measure data

Abstract.We give a partial uniqueness result concerning comparable renormalized solutions of the nonlinear elliptic problem -div(a(x,Du))=μ in Ω, u=0 on ∂Ω, where μ is a Radon measure with bounded variation on Ω.

HOMOGENIZATION OF AN ELLIPTIC SECOND-ORDER PROBLEM WITH L log L DATA IN A DOMAIN WITH OSCILLATING BOUNDARY

We consider a domain whose boundary presents numerous asperities. The asperities have fixed height, a size depending on a small parameter e and a e-periodic structure. We study the asymptotic behavior, as e vanishes, of a second-order elliptic Neumann problem in this domain with a data having L log La priori estimates. We identify the limit problem.

INFINITE VALUED SOLUTIONS OF NON-UNIFORMLY ELLIPTIC PROBLEMS

We consider a quasilinear equation (see (1.1)) with L1 data and with a diffusion matrix A(x,u), which is not uniformly coercive with respect to u (see Assumptions (H3)–(H4)). Under such assumptions it is not realistic, in general, to search for a solution which is finite almost everywhere. We introduce two equivalent notions of solutions which take into account the possible values +∞ and -∞ (see Definitions 2.1 and 2.3). Then we prove that there exists at least one such solution. At last we establish an uniqueness result in the class of simultaneous infinite valued solutions.

Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms

In this paper we prove uniqueness results for renormalized solutions to a class of nonlinear parabolic problems.

Symmetry and asymmetry of minimizers of a class of noncoercive functionals

Abstract In this paper we prove symmetry results for minimizers of a noncoercive functional defined on the class of Sobolev functions with zero mean value. We prove that the minimizers are foliated Schwarz symmetric, i.e., they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. In the two-dimensional case, we show a symmetry breaking.

Renormalized solutions of elliptic equations with Robin boundary conditions

Abstract In the present paper, we consider elliptic equations with nonlinear and nonhomogeneous Robin boundary conditions of the type { - div ( B ( x , u ) ∇ u ) = f in Ω , u = 0 on Γ o B ( x , u ) ∇ u ⋅ n → + γ ( x ) h ( u ) = g on Γ 1 where f and g are the element of L1 (Ω) and L1 (Γ1), respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additional assumptions on the matrix field B we show that the renormalized solution is unique.

Existence and uniqueness results for a nonlinear stationary system

AbstractWe prove a few existence results of a solution for a static system with a coupling of thermoviscoelastic type. As this system involves L¹ coupling terms we use the techniques of renormalized solutions for elliptic equations with L¹ data. We also prove partial uniqueness results.