The Method of Intrinsic Scaling
A Systematic Approach to Regularity for Degenerate and Singular PDEs
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Article
BMO-regularity for a degenerate transmission problem
We examine a transmission problem driven by a degenerate quasilinear operator with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis: the absence of represen...
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Article
On a two-phase free boundary problem ruled by the infinity Laplacian
In this paper we consider a two-phase free boundary problem ruled by the infinity Laplacian. Our main result states that bounded viscosity solutions in B1 are universally Lipschitz continuous in B1/2, which is th...
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Chapter and Conference Paper
Geometric Tangential Analysis and Sharp Regularity for Degenerate pdes
We provide a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto’s revolutionary method of intrinsic scaling, wh...
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Article
Sharp regularity for the inhomogeneous porous medium equation
We show that locally bounded solutions of the inhomogeneous porous medium equation $$u_{t}-\operatorname{div}\left(m|u|^{m-1} \nabla u\right)=f \in L^{q, r}, \quad m>1$$ are locally Hölder continuous, with expone...
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Article
On the interior regularity of weak solutions to the 2-D incompressible Euler equations
We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in develo...
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Article
Regularity for anisotropic fully nonlinear integro-differential equations
We consider fully nonlinear integro-differential equations governed by kernels that have different homogeneities in different directions. We prove a nonlocal version of the ABP estimate, a Harnack inequality a...
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Article
A Quantitative Modulus of Continuity for the Two-Phase Stefan Problem
We derive the quantitative modulus of continuity $$\omega(r)=\left[ p+\ln \left( \frac{r_0}{r}\right)\right]^{-\alpha (n, p)},$$ ...
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Article
An intrinsic Liouville theorem for degenerate parabolic equations
We show that weak solutions of the degenerate p−Laplace equation $$u_t - {\rm div}\left( |\nabla u|^{p-2}\nabla u \right) = 0,\quad p ...
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Article
Hölder continuity for Trudinger’s equation in measure spaces
We complete the study of the regularity for Trudinger’s equation by proving that weak solutions are Hölder continuous also in the singular case. The setting is that of a measure space with a doubling non-trivi...
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Article
The obstacle problem for nonlinear elliptic equations with variable growth and L 1-data
The aim of this paper is twofold: to prove, for L 1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth,...
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Book
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Article
Open AccessHarnack's Estimates: Positivity and Local Behavior of Degenerate and Singular Parabolic Equations
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Chapter and Conference Paper
Boundary Regularity at {t = 0} for a Singular Free Boundary Problem
In this note it is shown that the weak solutions of the Stefan problem for the singular p-Laplacian are continuous up to {t = 0}. The result is a follow-up to a recent paper of the authors concerning the interior...
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Book and Conference Proceedings
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Article
Uniqueness for nonlinear degenerate problems
We prove existence and uniqueness results for weak solutions of nonlinear degenerate problems arising in various physical models. The main novelty in the article concerns the uniqueness, which employs a techni...
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Chapter and Conference Paper
A Free Boundary Problem: Contributions from Modern Analysis
We exemplify the role of Free Boundary Problems as an important source of ideas in modern analysis. With the help of a model problem we illustrate the use of analytical, algebraic and geometrical techniques ob...
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Article
Continuous solutions for a degenerate free boundary problem
We prove existence of continuous solutions for $$\partial _t [\gamma \left( \theta \right)] - div(\left| {\nabla \theta } \right|^{p ...
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Article
On a Darcy–Stefan Problem arising in freezing and thawing of saturated porous media
A model with phase change for material convection in a saturated porous medium with a frozen region is formulated as a Darcy-Stefan problem. We propose a new generalized formulation for this Stefan-type probl...
