Related papers: Regularity for minimizers for functionals of doubl…

A new solution is proposed to the long-standing problem of describing the quantum phase of a harmonic oscillator. In terms of an'exponential phase operator', defined by a new 'polar decomposition' of the quantized amplitude of the…

In this paper, we extend the uniform regularity estimates obtained by M. Avellanda and F. Lin in the paper of Compactness methods in the theory of homogenization (Comm. Pure Appl. Math. 40(1987), no.6, 803-847) to the more general second…

We show the H\"older continuity of quasiminimizers of the energy functionals $\int f(x,u,\nabla u)\,dx$ with nonstandard growth under the general structure conditions $$ |z|^{p(x)} - b(x)|y|^{r(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} +…

Motivated by the construction of time-periodic solutions for the three-dimensional Landau-Lifshitz-Gilbert equation in the case of soft and small ferromagnetic particles, we investigate the regularity properties of minimizers of the…

We propose a regularization of the formal differential expression of order $m \geqslant 3$ $$ l(y) = i^my^{(m)}(t) + q(t)y(t), \,t \in (a, b), $$ applying quasi-derivatives. The distribution coefficient $q$ is supposed to have an…

We consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field $a$. Extending the work of the first author, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], who…

We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free…

Many functionals of interest in statistics and machine learning can be written as minimizers of expected loss functions. Such functionals are called $M$-estimands, and can be estimated by $M$-estimators -- minimizers of empirical average…

We study minimizers of the functional $$ \int_{B_1^+}|\nabla u|^2 x_n^a\,d x +2\int_{B_1'} (\lambda_+ u^++\lambda_- u^-)\,d x', $$ for $a\in(-1,1)$. The problem arises in connection with heat flow with control on the boundary. It can also…

We address a classical open question by H.Brezis and R.Ignat concerning the characterization of constant functions through double integrals that involve difference quotients. Our first result is a counterexample to the question in its full…

We first survey the known results on functional equations for the double zeta-function of Euler type and its various generalizations. Then we prove two new functional equations for double series of Euler-Hurwitz-Barnes type with complex…

$\sigma$-Brjuno functions were introduced in \cite{MaMoYo_06} as an interesting variant of the classical Brjuno function, where one substitutes the $\log$ singularity at $x=0$ with the power law divergence $x^{-1/\sigma},$ $(\sigma>0).$ As…

Let $(\mathbb{X} , d, \mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial \Omega ) $ and let $B_x$ be the…

We consider a small random perturbation of the energy functional $$ [u]^2_{H^s(\Lambda, R^d)} + \int_\Lambda W(u(x)) dx $$ for $s \in (0,1),$ where the non-local part $ [u]^2_{H^s(\Lambda,R^d)}$ denotes the total contribution from $\Lambda…

In this paper we study the regularity of the local minima of integral functionals: in particular, not convexity (quasi-convexity, policonvexity or rank one convexity) hypothesis will be made on the density, neither structure hypothesis nor…

We study the regularity of minimizers to the composite membrane problem in the plane (ie given a domain omega and a positive number A, smaller than the measure of omega, minimize the first Dirichlet eigenvalue for the Schrodinger operator…

We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the $H^1$ projection of measure-preserving maps. Our result introduces a new criteria on the…

We establish partial regularity results for minimizers of a class of functionals depending on differential expressions based on elliptic operators. Specifically, we focus on functionals of Orlicz growth with a natural strong quasiconvexity…

We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in $\mathbb{R}$. We prove that, after an appropriate translation, each of them is necessarily an even function which…

In this paper, we will use a suitable tranform to investigate the sharp constants and optimizers for the following Caffarelli-Kohn-Nirenberg inequalities for a wide range of parameters $(r,p,q,s,\mu,\sigma)$ and $0\leq a\leq1$:…