1,720,967 research outputs found
Logarithmic Sobolev trace inequalities
No full textWe prove a logarithmic Sobolev trace inequality in a gaussian space and we study the trace operator in the weighted Sobolev space W^{1,p}(\Omega,\gamma) for sufficiently regular domain. Applications to PDE are also considered
Quantitative isoperimetric inequalities for log-convex probability measures on the line
No full textThe purpose of this paper is to analyze the isoperimetric inequality for symmetric log-convex probability measures on the line. Using geometric arguments we first re-prove that extremal sets in the isoperimetric inequality are intervals or complement of intervals (a result due to Bobkov and Houdré). Then we give a quantitative form of the isoperimetric inequality, leading to a somehow anomalous behavior. Indeed, it could be that a set is very close to be optimal, in the sense that the isoperimetric inequality is almost an equality, but at the same time is very far (in the sense of the symmetric difference between sets) from any extremal sets! From the results on sets we derive quantitative functional inequalities of weak Cheeger type
Local Boundedness for Minimizers of Anisotropic Functionals with Monomial Weights
No full textWe study the local boundedness of minimizers of non uniformly elliptic integral functionals with a suitable anisotropic p,q- growth condition. More precisely, the growth condition of the integrand function f(x,∇u) from below involves different pi>1 powers of the partial derivatives of u and some monomial weights |xi|α[email protected]@6159b4dc with αi∈[0,1) that may degenerate to zero. Otherwise from above it is controlled by a q power of the modulus of the gradient of u with q≥maxipi and an unbounded weight μ(x). The main tool in the proof is an anisotropic Sobolev inequality with respect to the weights |xi|α[email protected]@3ee15ae7
Half-space Gaussian symmetrization: applications to semilinear elliptic problems
Full text linkWe consider a class of semilinear equations with an absorption nonlinear zero order term of power type, where elliptic condition is given in terms of Gauss measure. In the case of the superlinear equation we introduce a suitable definition of solutions in order to prove the existence and uniqueness of a solution in ℝN without growth restrictions at infinity. A comparison result in terms of the half-space Gaussian symmetrized problem is also proved. As an application, we give some estimates in measure of the growth of the solution near the boundary of its support for sublinear equations. Finally we generalize our results to problems with a nonlinear zero order term not necessary of power type
Sobolev anisotropic inequalities with monomial weights
Full text linkWe derive some anisotropic Sobolev inequalities in Rn with a monomial weight in the general setting of rearrangement invariant spaces. Our starting point is to obtain an integral oscillation inequality in multiplicative form
Uniqueness results for strongly monotone operators related to Gauss measure
Full text linkIn the present paper we prove some uniqueness results for weak solutions to a class of problems, whose prototype is
−div ((ε + |∇u|^2)^{(p−2)/2 ∇uφ) = f φ in Ω
u = 0 on ∂Ω,
where ε ≥ 0,1 < p < +∞, φ(x) is the density of the N-dimensional Gauss measure, Ω is an open subset of RN (N > 1) with Gauss measure less than one and datum f belongs to the natural dual space. When p ≤ 2 we obtain a uniqueness result for ε = 0. While for p > 2 we have to consider ε > 0 unless the sign of f is constant. Some counterexamples are given too
Comparison results for a linear elliptic equation with mixed boundary conditions
No full textIn this paper we study a linear elliptic equation having mixed boundary conditions, defined in a connected open set
Ω of R n.
We prove a comparison result with a suitable symmetrized'' Dirichlet problem which cannot be uniformly elliptic depending on the regularity of
∂ Ω. Regularity results for non-uniformly elliptic equations are also given
Existence results for a class of degenerate elliptic equations
No full textIn the present paper we prove existence results for a class of nonlinear elliptic equations
whose prototype is -div (|D u|^(p−2) Du φ( x) ) + |D u|^σ φ( x)= g φ ; where Ω is an open set, u=0
on \partial Ω; the function φ( x) =
(2π)^ (n/2) exp (−|x|2 /2) is the density of Gauss measure and g \in the weighted Lorentz-Zygmund space L^r (log L)^(-1/2) (φ,Ω), 1<r<p’. The results are sharp in this class of spaces
Linear elliptic equations related to Gauss measure
No full textIn this paper we study a Dirichlet problem relative to the equation Lu = g \phi - (f_i \phi)(x_i), where L is a linear elliptic operator with lower-order terms whose ellipticity condition is given in terms of the function \phi(x), the density of the Gaussian measure
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