Conveners
Degenerate PDEs and Free Boundary Problems: MS-1-1
- José Miguel Urbano (KAUST)
Degenerate PDEs and Free Boundary Problems: MS-1-2
- Damião J. Araujo (Universidade Federal da Paraíba)
Degenerate PDEs and Free Boundary Problems: MS-1-3
- José Miguel Urbano (KAUST)
Description
Organisers: Damiāo J. Araújo, Miguel Urbano
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Mark Allen (Brigham Young University)10/09/2026, 10:30Degenerate PDEs and Free Boundary Problems
We will discuss a parabolic PDE that arises in models for combustion and wildfire. Our main concern is the fire front: how it moves and what shape it can take. In the study of PDEs, the fire front is the free boundary. In this problem, at a fixed time the shape of the fire front can have singularities. However, when the fire front is viewed as a free boundary in space-time, we show that the...
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Hector Chang-Lara (Universidade de Coimbra and CIMAT)10/09/2026, 11:00Degenerate PDEs and Free Boundary Problems
Many equations arising in mathematics and the sciences have a remarkable smoothing effect. The heat equation provides a striking example: even when starting from very irregular initial data, its solutions become smooth instantaneously. Classical regularity theory seeks to explain this phenomenon for broad classes of elliptic and parabolic equations. In many important problems, however, the...
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Sunghan Kim (KTH Royal Institute of Technology)10/09/2026, 11:30Degenerate PDEs and Free Boundary Problems
Constraint maps are energy-minimizers under an image constraint, which is a natural vectorial extension of the obstacle problem. These maps develop free boundaries, due to the presence of an obstacle, while they also develop singularities, such as discontinuities, often for topological reasons. The partial regularity theory of these maps was established in the nineties, little has been known...
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Dr Hayk Mikayelyan (University of Nottingham Ningbo China)10/09/2026, 12:00Degenerate PDEs and Free Boundary Problems
Consider the cylindrical domain $\Omega=D\times(0,1)$ and the convex functional
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$$ \int_\Omega \frac{1}{2}|\nabla U(x)|^2dx +\int_D V(x')^+\,dx', $$ with nonlocal obstacle acting on function $V(x')=\int_0^1 U(x', t) dt $. We show that the unique minimizer solves the equation $$
\Delta U(x',x_n) = \chi_{{V>0}}(x') + \chi_{{V=0}}(x') [\partial_\nu U (x',0) + \partial_\nu U... -
Cintia Pacchiano (Universidad Nacional Autónoma de México)10/09/2026, 14:00Degenerate PDEs and Free Boundary Problems
In this talk, we present a unified regularity framework for variational integrals with non-uniformly elliptic integrands, including those exhibiting $p,q$-growth or exponential-type growth. We consider general energy functionals of the form
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$$ \int_{\Omega} f(x, Du) \, dx, $$ where the integrand $f(x, \xi)$ may satisfy natural growth, $(p,q)$-growth, or exponential growth conditions. We... -
Edgard Pimentel (CMUC, University of Coimbra)10/09/2026, 14:30Degenerate PDEs and Free Boundary Problems
Fully nonlinear free transmission problems have been examined from a number of perspectives. These include the existence of solutions, regularity estimates and numerical methods. However, the uniqueness of solutions remains fairly open. We resort to a relaxed notion of viscosity solutions and relate it with the usual definition. We also impose conditions on the operators driving the free...
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Dr Disson dos Prazeres (Universidade Federal de Sergipe)10/09/2026, 15:00Degenerate PDEs and Free Boundary Problems
In this talk, we will discuss conditions for the existence of solutions exhibiting a dead core in nonlocal problems. In particular, we will analyze conditions that ensure the formation of a dead core, as well as existence arguments and examples illustrating this phenomenon in the context of nonlocal partial differential equations.
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Dr Rafayel Teymurazyan (KAUST)10/09/2026, 15:30Degenerate PDEs and Free Boundary Problems
We present recent advances on the degenerate quenching problem. Our main result establishes the local finiteness of the (n-1)-dimensional Hausdorff measure of the free boundary. The proof combines optimal gradient decay estimates, derived from an intrinsic Harnack-type inequality, with a detailed analysis of the flatness regime, where minimizers enjoy improved regularity. This approach yields...
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Aelson Sobral (King Abdullah University of Science and Technology)11/09/2026, 09:00Degenerate PDEs and Free Boundary Problems
We develop a unified variational framework for singular energies that arise as analytic models in Simon's symmetric minimal surface program. Motivated by the linearized profile $\Delta u \sim u^{-1}$, we study local minimizers of energies whose potential $\sigma$ may diverge at the origin. This class encompasses both the logarithmic potentials linked to the geometry of minimal cones and the...
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María Soria Carro (Universidad Autónoma de Madrid)11/09/2026, 09:30Degenerate PDEs and Free Boundary Problems
Segregation problems arise when competing species tend to occupy disjoint regions, leading naturally to a free boundary separating them. In this talk, we will discuss a parabolic model in which the diffusion is governed by fully nonlinear operators. We will describe the main ideas behind the existence of Lipschitz solutions, the free boundary condition, and the regularity of the interface near...
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Marvin Weidner (University of Bonn)11/09/2026, 10:00Degenerate PDEs and Free Boundary Problems
A classical result by Kinderlehrer-Nirenberg states that C^{1,\alpha} regularity of the free boundary near a regular point implies that the free boundary is smooth. This principle applies to a wide range of free boundary problems for second-order operators. The goal of this talk is to explore counterparts of such higher regularity results for nonlocal free boundary problems, focusing on the...
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