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Raffaella Giova

Publications and source records attributed to Raffaella Giova.

10 recordsLinked to original sources
arXiv2026

Almost Lipschitz regularity for solutions of elliptic equations with discontinuous coefficients

We are interested in the local higher integrability of solutions to elliptic equations with linear growth of the form $$-\text{ div}A(x,Du)=f(x). $$ Under a Besov regularity assumption both on the partial map $x \mapsto A(x,ξ)$ and the datum $f$, we prove that the solutions are almost Lipschitz continuous, i.e. their gradients belong locally to $L^q$, for any finite exponent $q$. In turn, solutions are locally $γ$-Hölder continuous, for every $γ\in (0,1)$. The difficulty arising from the lack of an explicit second variation for the problem is overcome by testing the equation with a function proportional to a power of the finite difference quotient of the solution. To the best of our knowledge, this technique is used in this context for the first time. We also provide an example showing the sharpness of our result in the scale of Lebesgue spaces.

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arXiv2024

Gradient regularity for a class of elliptic obstacle problems

We prove some regularity results for a priori bounded local minimizers of non-autonomous integral functionals of the form $$\mathcal{F}(v,Ω)=\int_ΩF(x,Dv)dx,$$ under the constraint $v \ge ψ$ a.e. in $Ω$, where $ψ$ is a fixed obstacle function. Assuming that the coefficients of the partial map $x \mapsto D_ξF(x,ξ)$ satisfy a suitable Sobolev regularity, we are able to obtain higher differentiability and Lipschitz continuity results for the local minimizers.

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arXiv2022

Regularity results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

We establish some higher differentiability results for solution to non-autonomous obstacle problems of the form \begin{equation*} \min \left\{\int_Ωf\left(x, Dv(x)\right)dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\}, \end{equation*} where the function $f$ satisfies $p-$growth conditions with respect to the gradient variable, for $1<p<2$, and $\mathcal{K}_ψ(Ω)$ is the class of admissible functions. Here we show that, if the obstacle $ψ$ is bounded, then a Sobolev regularity assumption on the gradient of the obstacle $ψ$ transfers to the gradient of the solution, provided the partial map $x\mapsto D_ξf(x,ξ)$ belongs to a Sobolev space, $W^{1, p+2}$. The novelty here is that we deal with subquadratic growth conditions with respect to the gradient variable, i.e. $f(x, ξ)\approx a(x)|ξ|^p$ with $1<p<2,$ and where the map $a$ belongs to a Sobolev space.

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arXiv2021

Regularity results for bounded solutions to obstacle problems with non-standard growth conditions

In this paper we consider a class of obstacle problems of the type %\begin{equation*} %\int_Ω\left \, \dx\ge0\qquad\forall %φ\in W^{1,q}(Ω) \quad {\mathrm{s.t.}} \quad φ\ge ψ%\end{equation*} \begin{equation*} \min \left\{\int_Ωf(x, Dv)\, \dx\,:\, v\in \mathcal{K}_ψ(Ω)\right\} \end{equation*} where $ψ$ is the obstacle, $\mathcal{K}_ψ(Ω)=\{v\in u_0+W^{1, p}_{0}(Ω, \R): v\geψ\text{ a.e. in }Ω\}$, with $u_0 \in W^{1,p}(Ω)$ a fixed boundary datum, the class of the admissible functions and the integrand $f(x, Dv)$ satisfies non standard $(p,q)$-growth conditions. \\ We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map $x\mapsto A(x, ξ)$ is independent of the dimension $n$ and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one $W^{1,n}$.

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arXiv2020

Elliptic Equations With Degenerate weights

We obtain new local Calderon-Zygmund estimates for elliptic equations with matrix-valued weights for linear as well as non-linear equations. We introduce a novel log-BMO condition on the weight M. In particular, we assume smallness of the logarithm of the matrix-valued weight in BMO. This allows to include degenerate, discontinuous weights. We provide examples that show the sharpness of the estimates in terms of the log-BMO-norm.

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arXiv2019

Regularity results for solutions to obstacle problems with Sobolev coeffcients

We establish the higher differentiability of solutions to a class of obstacle problems for integral functionals where the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned , under a suitable Sobolev assumption on the partial map that measures the oscillation of f with respect to the x variable. The main novelty is that such assumption is independent of the dimension n and that, in the case p<=n-2, improves previous known results.

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arXiv2019

Very degenerate elliptic equations under almost critical Sobolev regularity

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.

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arXiv2016

Fractional differentiability for solutions of nonlinear elliptic equations

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,ξ)$ enjoys $B^α_{\frac{n}α, q}$ smoothness, local well-posedness is found in $B^α_{p,q}$ for certain values of $p\in[2,\frac{n}α)$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,ξ)=A(x)ξ$, $G=0$ and $A\in B^α_{\frac{n}α,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^α_{p,q}$ for each $p<\frac{n}α$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,ξ)$ satisfies a locally uniform $VMO$ condition.

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arXiv2008

A sharp Wirtinger inequality and some related functional spaces

We consider the generalized Wirtinger inequality \[ (\int_{0}^{T} a |u|^q )^{1/q} \le C \biggm(\int_{0}^{T} a^{1-p} |u'|^{p}\biggm)^{1/p}, \] with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge0$, $a\not\equiv0$ and where $u$ is a $T$-periodic function satisfying the constraint \[ \int_{0}^{T} a |u|^{q-2}u =0. \] We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined.

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