Related papers: $\mathbb{C}$-elliptic operators and $\mathrm{W}^{1…

We study $H^1$ versus $C^1$ local minimizers for functionals defined on spaces of symmetric functions, namely functions that are invariant by the action of some subgroups of $\mathcal{O}(N)$. These functionals, in many cases, are associated…

We study the regularity properties of H\"older continuous minimizers to non-autonomous functionals satisfying $(p,q)$-growth conditions, under Besov assumptions on the coefficients. In particular, we are able to prove higher integrability…

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…

We investigate fractional regularity estimates up to the boundary for solutions to fully nonlinear elliptic equations with measurable ingredients. Specifically, under the assumption of uniform ellipticity of the operator, we demonstrate…

We announce new existence and $\varepsilon$-regularity results for minimisers of the relaxation of strongly quasiconvex integrals that on smooth maps $u\colon\Omega\subset\mathbb{R}^{n}\to\mathbb{R}^{N}$ are defined by $$u\mapsto…

We consider an elliptic Kolmogorov equation $\lambda u - Ku = f$ in a separable Hilbert space $H$. The Kolmogorov operator $K$ is associated to an infinite dimensional convex gradient system: $dX = (AX - DU(X))dt + dW (t)$, where $A $ is a…

We establish the first partial regularity result for local minima of strongly $\mathscr{A}$-quasiconvex integrals in the case where the differential operator $\mathscr{A}$ possesses an elliptic potential $\mathbb{A}$. As the main…

We prove the higher differentiability of integer order of locally bounded minimizers of integral functionals of the form \begin{equation*} \mathcal{F}(u,\Omega):= \,\sum_{i=1}^{n} \dfrac{1}{p_i}\displaystyle \int_\Omega \, a_i(x) \lvert…

We establish local higher integrability and differentiability results for minimizers of variational integrals $$ \mathfrak{F}(v,\Omega) = \int_{\Omega} /! F(Dv(x)) \, dx $$ over $W^{1,p}$--Sobolev mappings $u \colon \Omega \subset {\mathbb…

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps $W^{1,p}(\mathbb{R}) \times W^{1,q}(\mathbb{R}) \to W^{1,r}(\mathbb{R})$ with $1 <p,q < \infty$ and $r\geq 1$, boundedly and…

We present and prove a version of the elliptic regularity theorem for partial differential equations involving fractional Riemann-Liouville derivatives. In this case, regularity is defined in terms of Sobolev spaces $H^s(X)$: if the forcing…

We consider a class of convex integral functionals composed of a term of linear growth in the gradient of the argument, and a fidelity term involving $L^2$ distance from a datum. Such functionals are known to attain their infima in the $BV$…

In this work, we consider a class of second order uniformly elliptic operators with smooth and bounded coefficients. We provide some estimates on the norm of the semigroup generated by these operators acting on weighted Sobolev spaces,…

We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump L\`evy processes. If the index of the…

We prove an a priori estimate for the second derivatives of local minimizers of integral functionals of calculus of variation with convex integrand with respect to the gradient variable, assuming that the function that measures the…

Sobolev type inequalities involving homogeneous elliptic canceling differential operators and rearrangement-invariant norms on the Euclidean space are considered. They are characterized via considerably simpler one-dimensional Hardy type…

A partial regularity theorem is presented for minimisers of $k$th-order functionals subject to a quasiconvexity and general growth condition. We will assume a natural growth condition governed by an $N$-function satisfying the $\Delta_2$…

We establish optimal $C^s$ boundary regularity for the most general class of (linear and translation invariant) nonlocal elliptic operator of order $2s$. Namely, we consider L\'evy operators that are symmetric and its Fourier symbol…

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…

We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with…