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Giuseppe Mingione

Publications and source records attributed to Giuseppe Mingione.

At least 19 recordsLinked to original sources
arXiv2026

Partial regularity in nonlocal systems I

Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy thresholds for nonlocal excess functionals. The analysis is carried out via the use of nonlinear potentials and allows to derive fine properties of solutions under sharp assumptions on data and kernel coefficients.

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arXiv2026

Partial regularity in nonlocal systems II

Solutions to nonlinear nonlocal systems of order $2s>1$ in $\mathbb{R}^n$ are $C^{1,α}$, for every $α<2s-1$, outside a closed singular set whose Hausdorff dimension is less than $n-2$, and which is empty when $n=2$.

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arXiv2026

Coarse-grained ellipticity and De Giorgi-Nash-Moser theory

We prove local boundedness and a Harnack inequality for nonnegative weak solutions of the equation $-\nabla\cdot(\mathbf{a}(x)\nabla u)=0$ under a coarse-grained ellipticity assumption on the symmetric coefficient field $\mathbf{a}$. Coarse-grained ellipticity is a scale-dependent condition, defined for fields with only $\mathbf{a},\mathbf{a}^{-1}\in L^1$, in terms of families of effective diffusion matrices on triadic cubes of all sizes, and our estimates depend quantitatively on a corresponding coarse-grained ellipticity ratio. We show that coarse-grained ellipticity can be enforced by purely negative Sobolev regularity hypotheses: if $\mathbf{a}\in L^1\cap W^{-s,p}(U)$ and $\mathbf{a}^{-1}\in L^1\cap W^{-t,q}(U)$ for exponents $p,q\in[1,\infty]$ and $s,t\in[0,1)$ satisfying $s<1-\frac{1}{p}$, $t<1-\frac{1}{q}$ and \[ \frac{s+t}{2} + \frac{d}{2}\Bigl(\frac{1}{p}+\frac{1}{q}\Bigr) < 1, \] then $\mathbf{a}$ is coarse-grained elliptic in $U$ and every nonnegative solution satisfies a quantitative unit-scale Harnack inequality. In particular, when $s=t=0$ we recover Trudinger's classical result under the integrability condition $\mathbf{a}\in L^p$, $\mathbf{a}^{-1}\in L^q$ with $\frac{1}{p}+\frac{1}{q}<\frac{2}{d}$, and we obtain the sharp scaling of the Harnack constant in terms of $\|\mathbf{a}\|_{L^p}$ and $\|\mathbf{a}^{-1}\|_{L^q}$. More importantly, our criteria apply to new classes of degenerate and singular coefficient fields for which $\mathbf{a},\mathbf{a}^{-1}\notin L^{1+δ}$ for all $δ>0$, including examples generated by singular fractal measures and Gaussian multiplicative chaos, beyond the reach of previous approaches based solely on integrability assumptions.

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arXiv2024

Nonlinear potential theoretic methods in nonuniformly ellliptic problems

Nonuniform ellipticity is a classical topic in the theory of partial differential equations. While several results in regularity theory have been adding up over decades, many basic issues, as for instance the validity of Schauder theory and sharp dependence of regularity upon data, remained opened for a while. In these notes we give an overview of recent results and techniques about the topic, that, via a novel use of nonlinear potential theoretic methods, allow to answer several of the above questions.

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arXiv2023

Regularity for double phase problems at nearly linear growth

Minima of functionals of the type $$ w\mapsto \int_Ω\left[\snr{Dw}\log(1+\snr{Dw})+a(x)\snr{Dw}^{q}\right] \dx\,, \quad 0\leq a(\cdot) \in C^{0, α}\,,$$ with $Ω\subset \er^n$, have locally Hölder continuous gradient provided $1 < q < 1+α/n$.

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arXiv2022

Nonuniformly elliptic Schauder theory

Local Schauder estimates hold in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic variational problems and elliptic equations are locally Hölder continuous, provided coefficients are locally Hölder continuous.

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arXiv2021

Lipschitz bounds and nonautonomous integrals

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and yield new, optimal regularity criteria even in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.

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arXiv2021

Interpolative gap bounds for nonautonomous integrals

For nonautonomous, nonuniformly elliptic integrals with so-called $(p,q)$-growth conditions, we show a general interpolation property allowing to get basic higher integrability results for Hölder continuous minimizers under improved bounds for the gap $q/p$. For this we introduce a new method, based on approximating the original, local functional, with mixed local/nonlocal ones, and allowing for suitable estimates in fractional Sobolev spaces.

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arXiv2019

On the regularity of minima of non-autonomous functionals

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-energy conditions are satisfied. These are related to the occurrence of the so-called Lavrentiev phenomenon that that non-autonomous functionals might exhibit, and which is a natural obstruction to regularity. In the case of vector valued problems we concentrate on higher gradient integrability of minima. Instead, in the scalar case, we prove local Lipschitz estimates. We also present an approach via a variant of Moser's iteration technique that allows to reduce the analysis of several non-uniformly elliptic problems to that for uniformly elliptic ones.

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arXiv2019

Manifold constrained non-uniformly elliptic problems

We consider the problem of minimizing variational integrals defined on \cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand exhibiting different polynomial growth conditions and no homogeneity. We develop a few intrinsic methods aimed at proving partial regularity of minima and providing techniques for treating larger classes of similar constrained non-uniformly elliptic variational problems. In order to give estimates for the singular sets we use a general family of Hausdorff type measures following the local geometry of the integrand. A suitable comparison is provided with respect to the naturally associated capacities.

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arXiv2018

Optimal Lipschitz criteria and local estimates for non-uniformly elliptic problems

We report on new techniques and results in the regularity theory of general non-uniformly elliptic variational integrals. By means of a new potential theoretic approach we reproduce, in the non-uniformly elliptic setting, the optimal criteria for Lipschitz continuity known in the uniformly elliptic one and provide a unified approach between non-uniformly and uniformly elliptic problems.

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arXiv2017

Regularity for general functionals with double phase

We prove sharp regularity results for a general class of functionals of the type $$ w \mapsto \int F(x, w, Dw) \, dx\;, $$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$ w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;, $$ with $0<ν\leq b(\cdot)\leq L $. This changes its ellipticity rate according to the geometry of the level set $\{a(x)=0\}$ of the modulating coefficient $a(\cdot)$. We also present new methods and proofs, that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.

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arXiv2014

Borderline gradient continuity of minima

The gradient of any local minimiser of functionals of the type $$ w \mapsto \int_Ωf(x,w,Dw)\,dx+\int_Ωwμ\,dx, $$ where $f$ has $p$-growth, $p>1$, and $Ω\subset \mathbb R^n$, is continuous provided the optimal Lorentz space condition $μ\in L(n,1)$ is satisfied and $x\to f(x, \cdot)$ is suitably Dini-continuous.

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