Related papers: Geometric measure theory and differential inclusio…
The local symplectic theory of integrable systems is fundamental to understand their global theory, as well as the behavior near singularities of fundamental models from classical and quantum mechanics which are known to be integrable, such…
We consider integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![\Gamma]\!]$, where $Q\in \mathbb N \setminus \{0\}$ and $\Gamma$ is sufficiently smooth. We prove that, if $q\in \Gamma$…
Wiener's criterion for the regularity of a boundary point with respect to the Dirichlet problem for the Laplace equation has been extended to various classes of elliptic and parabolic partial differential equations. They include linear…
This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the…
We study semilinear rough stochastic partial differential equations as introduced in [Gerasimovi{\v{c}}s, Hairer; EJP 2019]. We provide $\mathcal{L}^p(\Omega)$-integrable a priori bounds for the solution and its linearization in case the…
A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. Following this practice, we demonstrate an analogue of a result of L. Caporaso, J.…
We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately…
In this paper we prove sharp regularity for a differential inclusion into a set $K\subset\mathbb{R}^{2\times 2}$ that arises in connection with the Aviles-Giga functional. The set $K$ is not elliptic, and in that sense our main result goes…
We focus our attention on the de Rham operators' underlying properties which are specified by intrinsic effects of differential geometry structures. And then we apply the procedure of regularization in the context of Lipschitz version of de…
We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a…
In the first part of the paper, we classify linear integrable (multi-dimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb C$, enjoying natural symmetries and the property that the restriction of their…
We study the classical spaces $L_{p}$ and $\ell_{p}$ for the whole range $0<p<\infty$ from a metric viewpoint and give a complete Lipschitz embeddability roadmap between any two of those spaces when equipped with both their ad-hoc distances…
Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on…
As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of $\mathbb{R}^r\times \mathbb{C}^n$ (for some $r$…
We obtain global and local theorems on the existence of invariant manifolds for perturbations of non autonomous linear difference equations assuming a very general form of dichotomic behavior for the linear equation. The results obtained…
In this paper we propose new insights and ideas to set up quantitative boundary estimates for solutions to Dirichlet problem of a class of fully non-linear elliptic equations on compact Hermitian manifolds with real analytic Levi flat…
We deal with Calder\'on's problem in a layered anisotropic medium $\Omega\subset\mathbb{R}^n$, $n\geq 3$, with complex anisotropic admittivity $\sigma=\gamma A$, where $A$ is a known Lipschitz matrix-valued function. We assume that the…
We consider the Dirichlet problem for solutions to general second-order homogeneous elliptic equations with constant complex coefficients. We prove that any Jordan domain with $C^{1,\alpha}$-smooth boundary, $0<\alpha<1$, is not regular…
In this paper, we are concerned with elliptic equations of $p$-Laplace type with measure data, which is given by $-div\big(a(x)(|\nabla u|^2+s^2)^{\frac{p-2}{2}}\nabla u\big)=\mu$ with $p>1$ and $s\geq0$. Under the assumption that the…
The geometric theory of additive separation of variables is applied to the search for multiplicative separated solutions of the bi-Helmholtz equation. It is shown that the equation does not admit regular separation in any coordinate system…