arXiv subjects

Pasquale Ambrosio

Publications and source records attributed to Pasquale Ambrosio.

13 recordsLinked to original sources

arXiv2026

Boundedness and contractive estimates for orthotropic, widely degenerate, doubly nonlinear diffusion equations

We study the regularity of weak solutions to doubly nonlinear orthotropic evolution equations of the form \[ \partial_{t}(\vert u\vert^{α-1}u)-\sum_{i=1}^{N}\partial_{i}\left[a_{i}(x,t)\,(|\partial_{i}u|-δ_{i})_{+}^{p-1}\frac{\partial_{i}u}{\vert\partial_{i}u\vert}\right]=f\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}=Ω\times(0,T), \] where $Ω$ is a bounded open subset of $\mathbb{R}^{N}$ for $N\geq2$, the coefficients $a_{i}$ are measurable and bounded, $α>0$, $p\in(1,\infty)$ and $δ_{1},\ldots,δ_{N}$ are non-negative numbers. We show that weak solutions are locally bounded due to their membership in a suitable De Giorgi-type energy class. We also obtain contractive estimates and global boundedness in space for solutions to a Cauchy problem associated with the above PDE. Our analysis extends analogous results available in the literature for diffusion equations that either do not exhibit double nonlinearity or are less degenerate than those considered here. Another main novelty of this paper is the presence of a source term $f$ on the right-hand side of the equation, for which we impose suitable integrability assumptions in the space-time variables.

math.AP↗

arXiv2026

Gradient bounds for a widely degenerate orthotropic parabolic equation

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-δ_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω\times I, \] where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $I\subset\mathbb{R}$ is a bounded open interval, $p\geq2$, $δ_{1},\ldots,δ_{n}$ are non-negative numbers and $\left(\,\cdot\,\right)_{+}$ denotes the positive part. We prove that the local weak solutions are locally Lipschitz continuous in the spatial variable. The main novelty here is that the above equation combines an orthotropic structure with a strongly degenerate behavior. We emphasize that our result can be considered, on the one hand, as the parabolic counterpart of the elliptic result established in [12], and on the other hand as an extension to a significantly more degenerate framework of the findings contained in [13].

math.AP↗

arXiv2026

Widely degenerate anisotropic diffusion: local boundedness and semicontinuity

We investigate the regularity of local weak solutions to evolution equations of the form \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\,\partial_{x_{i}}\left[a_{i}(x,t)\,(\vert\partial_{x_{i}}u\vert-δ_{i})_{+}^{p_{i}-1}\,\frac{\partial_{x_{i}}u}{\vert\partial_{x_{i}}u\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}\,=\,Ω\times(0,T)\,, \] where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, the coefficients $a_{i}$ are measurable and bounded, $p_{i}>1$ and $δ_{i}\geq0$ are fixed parameters. Under suitable assumptions on the exponents $p_{i}$, we first show that the local boundedness of weak solutions follows from their membership in an appropriate non-homogeneous parabolic De Giorgi class. We then establish the existence of semicontinuous representatives for local weak sub(super)-solutions. Our analysis extends analogous results available for less degenerate operators and generalizes the local boundedness results obtained in [7] to fully anisotropic, widely degenerate parabolic PDEs with non-smooth coefficients depending additionally on the space-time variables $(x,t)$, whose growth is governed by a family of exponents $p_{i}$ rather than by a single exponent.

math.AP↗

arXiv2026

Regularity of vectorial minimizers for non-uniformly elliptic anisotropic integrals

We establish the local boundedness of the local minimizers $u:Ω\rightarrow\mathbb{R}^{m}$ of non-uniformly elliptic integrals of the form $\int_Ωf(x,Dv)\,dx$, where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ ($n\geq2)$ and the integrand satisfies anisotropic growth conditions of the type \[ \sum_{i=1}^{n}λ_{i}(x)|ξ_{i}|^{p_{i}}\le f(x,ξ)\leμ(x)\left\{ 1+|ξ|^{q}\right\} \] for some exponents $q\geq p_{i}>1$ and with non-negative functions $λ_{i},μ$ fulfilling suitable summability assumptions. The main novelties here are the degenerate and anisotropic behaviour of the integrand and the fact that we also address the case of vectorial minimizers ($m>1$). Our proof is based on the celebrated Moser iteration technique and employs an embedding result for anisotropic Sobolev spaces.

math.AP↗

arXiv2025

Gradient regularity for strongly singular or degenerate elliptic and parabolic equations

We present recent advances in the regularity theory for weak solutions to some classes of elliptic and parabolic equations with strongly singular or degenerate structure. The equations under consideration satisfy standard $p$-growth and $p$-ellipticity conditions only outside a ball centered at the origin. In the elliptic setting, we describe Besov and Sobolev regularity results for suitable nonlinear functions of the gradient of the weak solutions, covering both the subquadratic ($1<p<2$) and superquadratic ($p\geq2$) regimes. Analogous results are obtained in the corresponding parabolic framework, where we address the higher spatial and temporal differentiability of the solutions under appropriate assumptions on the data.

math.AP↗

arXiv2025

Local boundedness for weak solutions to strongly degenerate orthotropic parabolic equations

We prove the local boundedness of local weak solutions to the parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-δ_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}=Ω\times(0,T]\,, \] where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, $p\geq2$, $δ_{1},\ldots,δ_{n}$ are non-negative numbers and $\left(\,\cdot\,\right)_{+}$ denotes the positive part. The main novelty here is that the above equation combines an orthotropic structure with a strongly degenerate behavior. The core result of this paper thus extends a classical boundedness theorem, originally proved for the parabolic $p$-Laplacian, to a widely degenerate anisotropic setting. As a byproduct, we also obtain the local boundedness of local weak solutions to the isotropic counterpart of the above equation.

math.AP↗

arXiv2025

On the second-order regularity of solutions to widely singular or degenerate elliptic equations

We consider local weak solutions to PDEs of the type \[ -\,\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\text{in}\,\,Ω, \] where $1 2$, and that $f\in L_{loc}^{{\frac{np}{n(p-1)+2-p}}}(Ω)$ if $1<p\leq2$. The conditions on the datum $f$ are essentially sharp. As a consequence, we obtain the local higher integrability of $Du$ under the same minimal assumptions on $f$. For $λ=0$, our results give back those contained in [12,28].

math.AP↗

arXiv2024

Sharp Sobolev regularity for widely degenerate parabolic equations

We consider local weak solutions to the widely degenerate parabolic PDE \[ \partial_{t}u-\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\qquad\mathrm{in}\ \ Ω_{T}=Ω\times(0,T), \] where $p\geq2$, $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $λ$ is a non-negative constant and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Assuming that the datum $f$ belongs to a suitable Lebesgue-Besov parabolic space when $p>2$ and that $f\in L_{loc}^{2}(Ω_{T})$ if $p=2$, we prove the Sobolev spatial regularity of a novel nonlinear function of the spatial gradient of the weak solutions. This result, in turn, implies the existence of the weak time derivative for the solutions of the evolutionary $p$-Poisson equation. The main novelty here is that $f$ only has a Besov or Lebesgue spatial regularity, unlike the previous work [6], where $f$ was assumed to possess a Sobolev spatial regularity of integer order. We emphasize that the results obtained here can be considered, on the one hand, as the parabolic analog of some elliptic results established in [5], and on the other hand as the extension to a strongly degenerate setting of some known results for less degenerate parabolic equations.

math.AP↗

arXiv2024

Gradient bounds for strongly singular or degenerate parabolic systems

We consider weak solutions $u:Ω_{T}\rightarrow\mathbb{R}^{N}$ to parabolic systems of the type \[ u_{t}-\mathrm{div}\,A(x,t,Du)=f \qquad \mathrm{in}\ Ω_{T}=Ω\times(0,T), \] where $Ω$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $T>0$ and the datum $f$ belongs to a suitable Orlicz space. The main novelty here is that the partial map $ξ\mapsto A(x,t,ξ)$ satisfies standard $p$-growth and ellipticity conditions for $p>1$ only outside the unit ball $\{\vertξ\vert<1\}$. For $p>\frac{2n}{n+2}$ we establish that any weak solution \[ u\in C^{0}((0,T);L^{2}(Ω,\mathbb{R}^{N}))\cap L^{p}(0,T;W^{1,p}(Ω,\mathbb{R}^{N})) \] admits a locally bounded spatial gradient $Du$. Moreover, assuming that $u$ is essentially bounded, we recover the same result in the case $1<p\leq\frac{2n}{n+2}$ and $f=0$. Finally, we also prove the uniqueness of weak solutions to a Cauchy-Dirichlet problem associated with the parabolic system above. We emphasize that our results include both the degenerate case $p\geq2$ and the singular case $1<p<2$.

math.AP↗

arXiv2024

A physics-informed deep learning approach for solving strongly degenerate parabolic problems

In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning frameworks for solving initial-boundary value problems involving nonlinear PDEs. Recently, PINNs have shown promising results in several application fields. Motivated by applications to gas filtration problems, here we present and evaluate a PINN-based approach to predict solutions to strongly degenerate parabolic problems with asymptotic structure of Laplacian type. To the best of our knowledge, this is one of the first papers demonstrating the efficacy of the PINN framework for solving such kind of problems. In particular, we estimate an appropriate approximation error for some test problems whose analytical solutions are fortunately known. The numerical experiments discussed include two and three-dimensional spatial domains, emphasizing the effectiveness of this approach in predicting accurate solutions.

math.NA↗

arXiv2023

Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation

We carry on the investigation started in [2] about the regularity of weak solutions to the strongly degenerate parabolic equation \[ u_{t}-\mathrm{div}\left[(\vert Du\vert-1)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right]=f\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,Ω_{T}=Ω\times(0,T), \] where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Here, we weaken the assumption on the right-hand side, by assuming that $f\in L_{loc}^{p'}\left(0,T;B_{p',\infty,loc}^α\left(Ω\right)\right)$, with $α\in(0,1)$ and $p'=p/(p-1)$. This leads us to obtain higher fractional differentiability results for a function of the spatial gradient $Du$ of the solutions. Moreover, we establish the higher summability of $Du$ with respect to the spatial variable. The main novelty of the above equation is that the structure function satisfies standard ellipticity and growth conditions only outside the unit ball centered at the origin. We would like to point out that the main result of this paper can be considered, on the one hand, as the parabolic counterpart of an elliptic result contained in [1], and on the other hand as the fractional version of some results established in [2].

math.AP↗

arXiv2023

Regularity results for a class of widely degenerate parabolic equations

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-ν)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in $Ω_{T}=Ω\times(0,T)$, where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $p\geq2$, $ν$ is a positive constant and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. Assuming that the datum $f$ belongs to a suitable Lebesgue-Sobolev parabolic space, we establish the Sobolev spatial regularity of a nonlinear function of the spatial gradient of the weak solutions, which in turn implies the existence of the weak time derivative $u_{t}$. The main novelty here is that the structure function of the above equation satisfies standard growth and ellipticity conditions only outside a ball with radius $ν$ centered at the origin. We would like to point out that the first result obtained here can be considered, on the one hand, as the parabolic counterpart of an elliptic result established in [5], and on the other hand as the extension to a strongly degenerate context of some known results for less degenerate parabolic equations.

math.AP↗

arXiv2021

Besov regularity for a class of singular or degenerate elliptic equations

Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE $-\mathrm{div}\left((\vert\nabla u\vert-1)_{+}^{q-1}\frac{\nabla u}{\vert\nabla u\vert}\right)=f$, $\mathrm{in}\,\,Ω$, where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ for $n\geq2$, $1<q<\infty$ and $\left(\,\cdot\,\right)_{+}$ stands for the positive part. We assume that the datum $f$ belongs to a suitable Sobolev or Besov space. The main novelty here is that we deal with the case of subquadratic growth, i.e. $1<q<2$, which has so far been neglected. In the latter case, we also prove the higher fractional differentiability of the solution to a variational problem, which is characterized by the above equation. For the sake of completeness, we finally give a Besov regularity result also in the case $q\geq2$.

math.AP↗