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We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies…

We prove local boundedness and H\"older continuity for weak solutions to nonlocal double phase problems concerning the following fractional energy functional \[ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|v(x)-v(y)|^p}{|x-y|^{n+sp}} +…

We consider vector valued weak solutions $u:\Omega_T\to \mathbb{R}^N$ with $N\in \mathbb{N}$ of degenerate or singular parabolic systems of type \begin{equation*} \partial_t u - \mathrm{div} \, a(z,u,Du) = 0 \qquad\text{in}\qquad \Omega_T=…

We prove existence of variational solutions for a class of nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u &+ \text{P.V.}\int_{\mathbb{R}^N}…

We establish regularity and, under suitable assumptions, convergence to stationary states for weak solutions of a parabolic equation with a non-linear non-local drift term; this equation was derived from a model of active Brownian particles…

We prove H\"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\Delta u-\frac{A}{|x|^{2+\beta}}u=0,\,\,(\beta\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01}…

The goal of this article is to establish local Lipschitz continuity of weak solutions for a class of degenerated elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity…

In this thesis, a unified approach to prove the boundedness of gradients of solutions to degenerate and singular elliptic and parabolic phi-Laplacian systems is presented. At first, a Cacciopoli-type energy inequality with an additional…

We study solutions ${\bf v}$ of the parabolic system of PDE $$ \partial_t\left(D\psi({\bf v})\right)=\text{div}DF(D{\bf v}). $$ Here $\psi$ and $F$ are convex functions, and this is a model equation for more general doubly nonlinear…

The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage…

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 study the existence, multiplicity and regularity results of weak solutions for the Dirichlet problem of a semi-linear elliptic equation driven by the mixture of the usual Laplacian and fractional Laplacian \begin{equation*} \left\{%…

We obtain local boundedness and maximum principles for weak subsolutions to certain infinitely degenerate elliptic divergence form equations, and the local boundedness turns out to be sharp in more than two dimensions, answering the `Moser…

We prove local boundedness, Harnack's inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable…

We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian $(-\Delta )^s$ for $s\in (\frac34,1)$, and we provide a new bootstrapping scheme that makes it possible to…

We study the regularity of weak solutions to a certain class of second order parabolic system under the only assumption of continuous coefficients. By using the $A-$caloric approximation argument, we claim that the weak solution $u$ to such…

We study the regularity and uniqueness of weak solutions of a degenerate parabolic equation, arising as the limit of a stochastic lattice model of self-propelled particles. The angle-average of the solution appears as a coefficient in the…

We consider the nonlocal double phase equation \begin{align*} \mathrm{P.V.} &\int_{\mathbb{R}^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))K_{sp}(x,y)\,dy\\ &+\mathrm{P.V.} \int_{\mathbb{R}^n} a(x,y)|u(x)-u(y)|^{q-2}(u(x)-u(y))K_{tq}(x,y)\,dy=0,…

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $\nabla \cdot (\lambda |\nabla u|^{p-2}\nabla u)=0$, where the variable coefficient $0\leq\lambda$ and its inverse…

We construct local (in time) strong solutions in {$H^s(\mathbb{R}^3)$, $s>3/2$} and global weak solutions with finite energy for both the Pauli-Darwin and the Pauli-Poisswell systems. These are the first rigorous results on local and global…