Faber-Krahn and Lieb-type inequalities for the composite membrane problem

The classical Faber-Krahn inequality states that, among all domains with given measure, the ball has the smallest first Dirichlet eigenvalue of the Laplacian. Another inequality related to the first eigenvalue of the Laplacian has been proved by Lieb in 1983 and it relates the first Dirichlet eigenvalues of the Laplacian of two different domains with the first Dirichlet eigenvalue of the intersection of translations of them. In this paper we prove the analogue of Faber-Krahn and Lieb inequalities for the composite membrane problem

ON THE HARMONIC CHARACTERIZATION OF DOMAINS VIA MEAN VALUE FORMULAS

The Euclidean ball have the following harmonic characterization, via Gauss-mean value property: Let D be an open set with finite Lebesgue measure and let x(0) be a point of D. If u(x(0)) = 1/|D| int_{D} u(y)dy for every nonnegative harmonic function u in D, then D is a Euclidean ball centered at x(0). On the other hand, on every sufficiently smooth domain D and for every point x(0) in D there exist Radon measures mu such that u(x(0)) = int_{D} u(y)d mu(y)for every nonnegative harmonic function u in D. In this paper we give sufficient conditions so that this last mean value property characterizes the domain D

On the equation det ∇u=f{{\rm det}\,\nabla{u}=f} with no sign hypothesis

We prove existence of $${u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}$$ satisfying $$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$$ where k≥1 is an integer, $${\Omega}$$ is a bounded smooth domain and $${f\in C^{k}(\overline{\Omega}) }$$ satisfies $$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$$ with no sign hypothesis on

Local boundedness for solutions of a class of nonlinear elliptic systems

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of m equations in divergence form, satisfying p growth from below and q growth from above, with p <= q; this case is known as p, q-growth conditions. Well known counterexamples, even in the simpler case p = q, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component u(alpha) of the solution u = (u(1),..., u(m)) satisfies an improved Caccioppoli's inequality and we get the boundedness of u(alpha) by applying De Giorgi's iteration method, provided the two exponents p and q are not too far apart. Let us remark that, in dimension n = 3 and when p = q, our result works for 3/2 < p <= 3, thus it complements the one of Bjorn whose technique allowed her to deal with p <= 2 only. In the final section, we provide applications of our result

Local boundedness of weak solutions to elliptic equations with p, q−growth

This article is dedicated to Giuseppe Mingione for his 50th birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the local boundedness of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under p, q-growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable

On the H\uf6lder continuity for a class of vectorial problems

In this paper we prove local H\uf6lder continuity of vectorial local minimizers of special classes of integral functionals with rank-one and polyconvex integrands. The energy densities satisfy suitable structure assumptions and may have neither radial nor quasi-diagonal structure. The regularity of minimizers is obtained by proving that each component stays in a suitable De Giorgi class and, from this, we conclude about the H\uf6lder continuity. In the final section, we provide some non-trivial applications of our results

Lipschitz regularity for degenerate elliptic integrals with p, q-growth

We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the x -variable

On the equation det del u = f with no sign hypothesis

We prove existence of u is an element of C-k ((Omega) over bar ;R-n) satisfyin