Related papers: Lower semicontinuity for an integral functional in…

We establish the higher fractional differentiability for the minimizers of non-autonomous integral functionals of the form \begin{equation} \mathcal{F}(u,\Omega):=\int_\Omega \left[ f(x,Du)- g \cdot u \right] dx , \notag \end{equation}…

We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. We show that the limits of these nonlocal functionals are…

This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{ and }\quad…

We study the regularity of Radon measures $\mu$ which satisfy that there exists a function $h_\mu$ in $H^1(\Omega)$, stationary harmonic such that $\Delta h_\mu =\mu$ in $\Omega$ (here $\Omega$ is an open set of $\mathbb{R}^2$). Such…

We establish partial regularity of BD-minima for variational integrals of linear growth which depend on the symmetric gradients and satisfy a weak ellipticity condition. Since there is no Korn Inequality in the $L^{1}$-Setup, the result…

Verifying lower-semicontinuity of integral functionals in the weak topology of Sobolev spaces is a central theme in the calculus of variations. For integral functionals with $p$-growth, quasiconvexity is a necessary condition for weak…

We prove partial regularity for minimizers of quasiconvex functionals of the type $\int_\Omega f(x,Du) dx$ with $p(x)$ growth with respect to the second variable. The proof is direct and uses a method of $A$-harmonic approximation.

We establish an $\varepsilon$-regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a consequence, the partial regularity of such BV…

Motivated by the testing condition for Radon-Brascamp-Lieb multilinear functionals established in arXiv:2201.12201, this paper is concerned with identifying local conditions on smooth maps $u(t)$ with values in the space of decomposable…

We investigate the negative part of the spectrum of the operator $-\partial^2 - \mu$ on $L^2(\mathbb R)$, where a locally finite Radon measure $\mu \geq 0$ is serving as a potential. We obtain estimates for the eigenvalue counting function,…

For a Radon measure $\mu$ on $\bbR,$ we show that $L^{\infty}(\mu)$ is invariant under the group of translation operators $T_t(f)(x) = {$f(x-t)$}\ (t \in \bbR)$ if and only if $\mu$ is equivalent to Lebesgue measure $m$. We also give…

For an arbitrary Radon measure $\mu$ we estimate the integrated discrete curvature of $\mu$ in terms of its centred variant of Jones' beta numbers. We farther relate integrals of centred and non-centred beta numbers. As a corollary,…

We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega, \] where $0 < q < 1$,…

Bounded weak solutions of Burgers' equation $\partial_tu+\partial_x(u^2/2)=0$ that are not entropy solutions need in general not be $BV$. Nevertheless it is known that solutions with finite entropy productions have a $BV$-like structure: a…

We study the existence of solutions of the nonlinear problem $$ \left\{ \begin{alignedat}{2} -\Delta u + g(u) & = \mu & & \quad \text{in } \Omega,\\ u & = 0 & & \quad \text{on } \partial \Omega, \end{alignedat} \right. $$ where $\mu$ is a…

We provide the integral representation formula for the relaxation in $BV(\Omega; \mathbb{R}^M)$ with respect to strong convergence in $L^1(\Omega; \mathbb{R}^M)$ of a functional with a boundary contact energy term. This characterization is…

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let $u$ be a real-valued harmonic function in $\mathbb{R}^n$ with $u(0)=0$ and $n\geq 3$. We prove…

We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega,\] for $0 < q < 1$, where $\sigma\ge 0$ is a measurable function, or a Radon…

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice…

We state necessary and sufficient conditions for weak lower semicontinuity of $u\mapsto\int_\Omega h(x,u(x))\,d x$ where $|h(x,s)|\le C(1+|s|^p)$ is continuous and possesses a recession function, and $u\in L^p(\Omega;\mathbb{R}^m)$, $p>1$,…