The behavior of the best Sobolev trace constant and extremals in thin domains

In this paper, we study the asymptotic behavior of the best Sobolev trace constant and extremals for the immersion W1,p(Ω)?Lq(∂Ω) in a bounded smooth domain when it is contracted in one direction. We find that the limit problem, when rescaled in a suitable way, is a Sobolev-type immersion in weighted spaces over a projection of Ω, W1,p(P(Ω),α)?Lq(P(Ω),β).For the special case p=q, this problem leads to an eigenvalue problem with a nonlinear boundary condition. We also study the convergence of the eigenvalues and eigenvectors in this case.     

Optimal boundary holes for the Sobolev trace constant

In this paper we study the problem of minimizing the Sobolev trace Rayleigh quotient among functions that vanish in a set contained on the boundary ∂Ω of given boundary measure.We prove existence of extremals for this problem, and analyze some particular cases where information about the location of the optimal boundary set can be given. Moreover, we further study the shape derivative of the Sobolev trace constant under regular perturbations of the boundary set.     

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Let Ω^Ω be the semigroup of all mappings of a countably infinite set Ω. If U and V are subsemigroups of Ω^Ω, then we write U≈V if there exists a finite subset F of Ω^Ω such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in Ω^Ω is the least cardinality of a subset A of Ω^Ω such that the union of U and A generates Ω^Ω. In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω.The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈. Moreover such semigroups have relative rank 0, 1, 2, or d in Ω^Ω where d is the minimum cardinality of a dominating family for N^N. We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in Ω^Ω are 0, 1, 2, and d.We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2^ℵ₀.     

Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u‖² _H ¹ _(Ω) ²/‖u‖² _L ² _(∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes. Mathematics Subject Classification (2000) 35P15, 49K20, 49M25, 49Q10     

A global Lipschitz continuity result for a domain-dependent Neumann eigenvalue problem for the Laplace operator

Let Ω be an open connected subset of Rⁿ of finite measure for which the Poincaré-Wirtinger inequality holds. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset φ(Ω) of Rⁿ, where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then, we show Lipschitz-type inequalities for the reciprocals of the eigenvalues delivered by the Rayleigh quotient. Then, we further assume that the imbedding of the Sobolev space W^1,2(Ω) into the space L²(Ω) is compact, and we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.     

Symmetry theorems for the overdetermined eigenvalue problems

The well-known Schiffer conjecture saying that for a smooth bounded domain Ω⊂Rn, if there exists a positive Neumann eigenvalue such that the corresponding Neumann eigenfunction u is constant on the boundary of Ω, then Ω is a ball. In this paper, we shall prove that the Schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding Neumann eigenfunction is constant on the boundary. We also prove a similar result to the Berenstein conjecture for the overdetermined Dirichlet eigenvalue problem.     

Best constant in critical Sobolev inequalities of second-order in the presence of symmetries

Let (M,g) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H²(M)?L^2?(M) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of (M,g). We also prove that we can take ?=0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz-Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation.     

Existence and non-existence of solutions for a class of Monge-Ampère equations

We study the boundary value problems for Monge-Ampère equations: detD²u=e−u in Ω⊂Rn, n?1, u_|∂Ω=0. First we prove that any solution on the ball is radially symmetric by the argument of moving plane. Then we show there exists a critical radius such that if the radius of a ball is smaller than this critical value there exists a solution, and vice versa. Using the comparison between domains we can prove that this phenomenon occurs for every domain. Finally we consider an equivalent problem with a parameter detD²u=e−tu in Ω, u_|∂Ω=0, t?0. By using Lyapunov-Schmidt reduction method we get the local structure of the solutions near a degenerate point; by Leray-Schauder degree theory, a priori estimate and bifurcation theory we get the global structure.     

The best Sobolev trace constant in a domain with oscillating boundary

In this paper we study homogenization problems for the best constant for the Sobolev trace embedding W^1,p(Ω)?L^q(∂Ω)$W^{1,p}\left(\Omega \right)?L^q\left(\partial \Omega \right)$ in a bounded smooth domain when the boundary is perturbed by adding an oscillation. We find that there exists a critical size of the amplitude of the oscillations for which the limit problem has a weight on the boundary. For sizes larger than critical the best trace constant goes to zero and for sizes smaller than critical it converges to the best constant in the domain without perturbations.     

Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ<γ^* then this stationary solution is unstable.     

Moser-Trudinger trace inequalities

Sharp constants in exponential inequalities involving a general class of measures in domains Ω⊂Rn are exhibited in the limiting case of the Sobolev embedding theorem. A comprehensive approach is presented yielding, as special instances, trace inequalities on ∂Ω, on smooth submanifolds of Ω of arbitrary dimension, and also on fractal subsets of Ω, and recovering, in particular, the classical Moser-Trudinger inequality.     

A Hardy inequality in the half-space

Here we prove a Hardy-type inequality in the upper half-space which generalize an inequality originally proved by Maz’ya (Sobolev Spaces, Springer, Berlin, 1985, p. 99). Here we present a different proof, which enable us to improve the constant in front of the remainder term. We will also generalize the inequality to the L^p case.     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

Sobolev spaces on an arbitrary metric measure space: Compactness of embeddings

We formulate a new definition of Sobolev function spaces on a domain of a metric space in which the doubling condition need not hold. The definition is equivalent to the classical definition in the case that the domain lies in a Euclidean space with the standard Lebesgue measure. The boundedness and compactness are examined of the embeddings of these Sobolev classes into L _q and C _α . We state and prove a compactness criterion for the family of functions L _p (U), where U is a subset of a metric space possibly not satisfying the doubling condition.     

Stochastic flows of SDEs with irregular coefficients and stochastic transport equations

In this article we study (possibly degenerate) stochastic differential equations (SDEs) with irregular (or discontinuous) coefficients, and prove that under certain conditions on the coefficients, there exists a unique almost everywhere stochastic (invertible) flow associated with the SDE in the sense of Lebesgue measure. In the case of constant diffusions and BV drifts, we obtain such a result by studying the related stochastic transport equation. In the case of non-constant diffusions and Sobolev drifts, we use a direct method. In particular, we extend the recent results on ODEs with non-smooth vector fields to SDEs.     

Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space

We consider the entire graph S of a continuous real function over RN−1 with N?3. Let Ω be a domain in RN with S as a boundary. Consider in Ω the heat flow with initial temperature 0 and boundary temperature 1. The problem we consider is to characterize S in such a way that there exists a stationary isothermic surface in Ω. We show that S must be a hyperplane under some general conditions on S. This is related to Liouville or Bernstein-type theorems for some elliptic Monge-Ampère-type equation.     

On symmetry of nonnegative solutions of elliptic equations

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H  . Moreover, we prove that if u?0$u?0$, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set.     

Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|^p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.     

Singular parabolic equations with measures as initial data

In this paper we study the Cauchy problem for the singular evolution p-Laplacian equations with gradient term and source on the assumption of measures as initial conditions. For the supercritical case q>p−1+p/N, we obtain that for every nonnegative solution there exists a nonnegative Radon measure μ as initial trace and μ has some local regularity.