Iñigo Urtiaga Erneta defended his PhD thesis Elliptic problems: regularity of stable solutions and a nonlocal Weierstrass extremal field theory on July 4, 2023.
The thesis was produced within the UPC doctoral program on Applied Mathematics and his thesis advisor was Xavier Cabré.
Currently, he is a Hill Assistant Professor at Rutgers University under the supervision of Yanyan Li.
Thesis summary
Partial Differential Equations (PDEs) are used to model almost every phenomenon affecting our daily lives, and they arise in areas as complex and diverse as physics, engineering, biology, or economics. Among these equations, elliptic PDEs describe stationary situations such as the equilibrium configurations of an evolution process. In applications, the interest lies in nonlinear equations which may admit too many different solutions. However, the only “physical” solutions one sees are the stable ones, namely, those that do not disappear under small perturbation of the data. Our thesis investigates qualitative properties of this natural class of solutions to elliptic problems.
The first part of the dissertation is devoted to the regularity of stable solutions to semilinear equations. This question is motivated by problems in combustion, where the temperature of a combustible mixture solves a reaction-diffusion equation and is expected to be near a stable solution to the associated elliptic problem. It has been known for a long time that these solutions may be singular when the dimension n (i.e., the number of variables in the problem) is sufficiently large. Namely, when n ⩾ 10, there are explicit examples of singular stable solutions where the “reaction term” (i.e., the nonlinearity of the problem) is an exponential function. Recently, in the breakthrough paper [3], X. Cabré, A. Figalli, X. Ros-Oton, and J. Serra showed that if n ⩽ 9, then stable solutions are smooth for any nonlinearity. Their proof applies to semilinear equations involving the Laplacian (an operator with constant coefficients) in a sufficiently regular domain. In [4–6] we extended their techniques to operators with variable coefficients, establishing the regularity of stable solutions in the same optimal range of dimensions. As a consequence of our analysis, we have even improved the known results for the Laplacian by significantly weakening the regularity requirements of the domain.
In the second part of the thesis, we develop an extremal field theory for nonlocal elliptic problems. Nonlocal equations (such as integro-differential equations) have gained much interest in recent years, as they are more suited than PDEs to model phenomena driven by long-range interactions. Many such equations arise from variational problems, where solutions can be interpreted as critical points (also known as “extremals”) of some energy functional. A fundamental question in the Calculus of Variations is to determine whether an extremal actually minimizes this energy. For classical, local problems, sufficient conditions for minimality have been known since the XIXth century. Most notably, we have the following remarkable result of Weierstrass: if a critical point is embedded in a family of extremals whose graphs produce a foliation (an “extremal field”), then it is a minimizer with respect to competitors taking values in the foliated region. To prove this theorem, one constructs a calibration for the energy, i.e., an auxiliary functional (satisfying appropriate technical conditions) which yields the minimality of the critical point as a direct consequence. Together with X. Cabré and J.C. Felipe-Navarro, in [2] we have extended the theory of extremal fields to the nonlocal setting for the first time. Our main achievement has been to construct a calibration for general nonlocal energy functionals. Before our work, calibrations had only been obtained for nonlocal minimal surfaces in [1]. To find a calibration for the Gagliardo seminorm (the most basic fractional functional) was an important open problem that we have solved.
Selected Publication: [2].
References
[1] X. Cabré, Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory, Ann. Mat. Pura Appl. 199 (2020), no. 5, 1979-1995.
[2] X. Cabré, I. U. Erneta, and J.C. Felipe-Navarro, A Weierstrass extremal field theory for the fractional Laplacian, Adv. Calc. Var. (2023). To appear. arXiv pdf (34 pages).
[3] X. Cabré, A. Figalli, X. Ros-Oton, and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math. 224 (2020), no. 2, 187–252.
[4] I. U. Erneta, Boundary Hölder continuity of stable solutions to semilinear elliptic problems in C1,1 domains. arXiv pdf (34 pages).
[5] _____, Energy estimate up to the boundary for stable solutions to semilinear elliptic problems, J. Differential Equations 378 (2024), 204–233. arXiv pdf.
[6] _____, Stable solutions to semilinear elliptic equations for operators with variable coefficients, Commun. Pure Appl. Anal. 22 (2023), 530–571. arXiv pdf.