Partial regularity for vectorial local-nonlocal problems
We establish sharp interior regularity for solutions to vectorial, nonlinear local-nonlocal systems.
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arXiv subjects
Paolo Baroni
Publications and source records attributed to Paolo Baroni.
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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Existence and boundary regularity for degenerate phase transitions
We study the Cauchy-Dirichlet problem associated to a phase transition modeled upon the degenerate two-phase Stefan problem. We prove that weak solutions are continuous up to the parabolic boundary and quantify the continuity by deriving a modulus. As a byproduct, these a priori regularity results are used to prove the existence of a so-called physical solution.
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The Cauchy-Dirichlet problem for a general class of parabolic equations
We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy-Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary $p$-Laplacian, but extending it at a wide scale. We employ a regularization technique of viscosity-type that we find interesting in itself.
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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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A quantitative modulus of continuity for the two-phase Stefan problem
We derive the quantitative modulus of continuity $$ ω(r)=\left[ p+\ln \left( \frac{r_0}{r} \right) \right]^{-α(n,p)}, $$ which we conjecture to be optimal, for solutions of the $p$-degenerate two-phase Stefan problem. Even in the classical case $p=2$, this represents a twofold improvement with respect to the 1984 state-of-the-art result by DiBenedetto and Friedman [J. reine angew. Math., 1984], in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent $α(n,p)$.
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Lorentz estimates for obstacle parabolic problems
We prove that the spatial gradient of (variational) solutions to parabolic obstacle problems of p-Laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of Lorentz spaces.
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Global estimates for nonlinear parabolic equations
We consider nonlinear parabolic equations of the type $$ u_t - div a(x, t, Du)= f(x,t) on Ω_T = Ω\times (-T,0), $$ under standard growth conditions on $a$, with $f$ only assumed to be integrable. We prove general decay estimates up to the boundary for level sets of the solutions $u$ and the gradient $Du$ which imply very general estimates in Lebesgue and Lorentz spaces. Assuming only that the involved domains satisfy a mild exterior capacity density condition, we provide global regularity results.
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