Related papers: Everywhere regularity results for a polyconvex fun…
The paper continues the analysis started in [Cora-Fioravanti-Vita-25,Fioravanti-24] on the local regularity theory for elliptic equations having coefficients which are degenerate or singular on some lower dimensional manifold. The model…
We introduce a weak asymptotic version of nonlinear contraction, termed \emph{asymptotic pointwise contraction}. For a mapping on a metric space, this notion requires the existence of a sequence of functions that dominate the distances…
We consider triplets of densities $(u_1,u_2,u_3)$ minimizing the Dirichlet energy \[\sum_{j=1}^3 \int_{\Omega} |\nabla u_j|^2\,dx \] over a bounded domain $\Omega\subset \mathbb{R}^N$, subject to the partial segregation condition: \[…
It is shown that any smooth strictly convex global solution of $$\det(\frac{\partial^{2}u}{\partial \xi_{i}\partial \xi_{j}}) = \exp \left\{-\sum_{i=1}^n d_i \frac{\partial u}{\partial \xi_{i}} - d_0\right\},$$ where $d_0$, $d_1$,...,$d_n$…
We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operator $$(\partial_t-\Delta)^su(t,x)=f(t,x),\quad\hbox{for}~0<s<1.$$ This nonlocal equation of order $s$ in time and…
In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work of Lopez-Rio imply rigidity of gradient shrinking Ricci solitons with harmonic Weyl…
In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl}…
We prove a sharp interior regularity theorem for fully nonlinear maximally subelliptic PDE in non-isotropic Sobolev spaces adapted to maximally subelliptic operators. This result complements the regularity theorem proven by Street for…
In large-scale applications, such as machine learning, it is desirable to design non-convex optimization algorithms with a high degree of parallelization. In this work, we study the adaptive complexity of finding a stationary point, which…
We establish the H\"older continuity of bounded nonnegative weak solutions to \begin{align*} \big(\Phi^{-1}(w)\big)_t=\Delta w+\nabla\cdot\big(a(x,t)\Phi^{-1}(w)\big)+b\big(x,t,\Phi^{-1}(w)\big), \end{align*} with convex $\Phi\in…
In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centered at that point to convex sets, even…
We investigate a class of composite nonconvex functions, where the outer function is the sum of univariate extended-real-valued convex functions and the inner function is the limit of difference-of-convex functions. A notable feature of…
Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $\Omega_T$. Fix a point $(x_o,t_o)\in\Omega_T$: if the average \[…
A set of locally finite perimeter $E \subset \mathbb{R}^{n}$ is called an anisotropic minimal surface in an open set $A$ if $\Phi(E;A) \le \Phi(F;A)$ for some surface energy $\Phi(E;A) = \int_{\partial^{*}E \cap A} \| \nu_{E}\| d…
We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order~$2$ in one variable. By constructing an explicit barrier, we…
In this paper, we consider a new class of multi phase operators with variable exponents, which reflects the inhomogeneous characteristics of hardness changes when multiple different materials are combined together. We at first deal with the…
This paper is a continuation of our previous work [21], where we have established that, for the second-order degenerate hyperbolic equation (\p_t^2-t^m\Delta_x)u=f(t,x,u), locally bounded, piecewise smooth solutions u(t,x) exist when the…
We establish universality of the renormalised energy for mappings from a planar domain to a compact manifold, by approximating subquadratic polar convex functionals of the form $\int_\Omega f(|\mathrm{D} u|)\,\mathrm{d} x$. The analysis…
We prove local regularity up to flat part of boundary, for certain classes of distributional solutions that are $L_{\infty}L^{3,q}$ with $q$ finite.
We establish partial H\"older regularity for (local) generalised minimisers of variational problems involving strongly quasi-convex integrands of linear growth, where the full gradient is replaced by a first order homogeneous differential…