1,720,991 research outputs found
Faber-Krahn and Lieb-type inequalities for the composite membrane problem
Full text linkThe 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
Hölder continuity of local minimizers of vectorial integral functional
No full textWe study the regularity of vector-valued local minimizers in W 1,p, p > 1, of the integral functional u → ∫ Ω [(μ2 + |Du|2)p/2 + f(x, u, |Du|)] dx, where Ω is an open set in RN and f is a continuous function, convex with respect to the last variable, such that 0 ≤ f(x, u, t) ≤ C(1 + tp). We prove that if f = f(x, t), or f = f(x, u, t) and p ≥ N, then local minimizers are locally Hölder continuous for any exponent less than 1. If f = f(x, u, t) and p < N then local minimizers are Hölder continuous for every exponent less than 1 in an open set Ω0 such that the Hausdorff dimension of Ω/Ω0 is less than N - p
On the Harmonic Characterization Of The Spheres: A Sharp Stability Inequality
No full textLet be a bounded open subset of with finite
-dimensional Hausdorff measure and let be a point of . We introduce a new harmonic invariant, that we call Kuran gap of w.r.t. . To define this new invariant, denoted , we use a family of harmonic functions introduced by \"Ulk\"u Kuran in \cite{kuran}. Our main stability result can be described as follows: if is sufficiently regular just in one of the points of nearest to , then is bounded from below by a kind of isoperimetric index, precisely the normalized difference between and , being the biggest ball contained in and centered at .
This partially extends and improves a stability result by Preiss and Toro.
By our stability result, we also obtain new rigidity results: (i) a characterization of the Euclidean spheres in terms of single-layer potentials, improving previous theorems by Fichera and by Shahgholian; (ii) a sufficient condition for a harmonic pseudosphere to be a Euclidean sphere, partially extending and improving rigidity results by Lewis and Vogel
On Mean Value formulas for solutions to second order linear PDEs
Full text linkIn this paper we give a general proof of Mean Value formulas for solutions to second order linear PDEs, only based on the local properties of their fundamental solution Gamma. Our proof requires a kind of pointwise vanishing integral condition for the intrinsic gradient of Gamma. Combining our Mean Value formulas with a "descent method" due to Kuptsov, we obtain formulas with improved kernels. As an application, we implement our general results to heat operators on stratified Lie groups and to Kolmogorov operators
Stability of the mean value formula for harmonic functions in Lebesgue spaces
Full text linkLet D be an open subset of Rn with finite measure, and let x∈ D. We introduce the p-Gauss gap of D w.r.t. x to measure how far are the averages over D of the harmonic functions u∈ Lp(D) from u(x). We estimate from below this gap in terms of the ball gap of D w.r.t. x, i.e., the normalized Lebesgue measure of D B, being B the biggest ball centered at x contained in D. From these stability estimates of the mean value formula for harmonic functions in Lp-spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space W1,p′, where p′ is the conjugate exponent of p
ON THE CHARACTERIZATION OF THE HARMONIC PSEUDOSPHERES VIA KURAN’S FUNCTIONS AND SINGLE-LAYER POTENTIALS
No full textWe present some characterizations of the harmonic pseudospheres in terms of the so called Kuran’s functions and of the single-layer potentials. Our characterizations apply to solid, harmonically stable domains
Hölder Continuity for Local Minimizers of a Nonconvex Variational Problem
No full textWe consider integral functional of the Calculus of Variations where the energy density is a continuous function with p-growth, p > 1, uniformly convex at infinity with respect to the gradient variable. We prove that local minimizers are α-Hölder continuous for all α < 1
Regularity of quasi-minimizers for non-uniformly elliptic integrals
Full text linkIn this paper we consider a class of non-uniformly elliptic integral functionals and we prove the local boundedness of the quasi-minimizers. Our approach is based on a suitable adaptation of the celebrated De Giorgi proof, and it relies on an appropriate Caccioppoli-type inequality
Local boundedness for solutions of a class of nonlinear elliptic systems
Full text linkIn 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
Regularity of minimizers of vectorial integrals with p - q growth
No full textThe regularity of minimizers of vectorial integrals with p - q growth is discussed. Local Lipschitz continuity of local minimizers of vectorial integrals is also proved. The uniform convexity and the radial structure condition with respect to the last variable are assumed only at infinity
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