arXiv subjects

Dimitrios Gazoulis

Publications and source records attributed to Dimitrios Gazoulis.

9 recordsLinked to original sources

arXiv2025

Minimality of level sets in phase transitions

In this work we study the level sets of entire solutions of the Allen-Cahn equation and we prove minimality of the zero level set with respect to a certain perimeter functional with density. This provides a direct relationship between phase transition type problems and minimal surfaces with some weight. In addition, we obtain that the zero level set of entire solutions of the Allen-Cahn equation has zero mean curvature. As an application, we establish the De Giorgi conjecture in it's original statement, without the limiting assumption of O. Savin, by reducing it to classical Bernstein type results for minimal graphs, thus directly linking it to the geometric problem.

math.AP↗

arXiv2025

On the Stability and Convergence of Physics Informed Neural Networks

Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering problems. The mathematical understanding of these methods is limited, and in particular, it seems that, a consistent notion of stability is missing. Towards addressing this issue we consider model problems of partial differential equations, namely linear elliptic and parabolic PDEs. Motivated by tools of nonlinear calculus of variations we systematically show that coercivity of the energies and associated compactness provide a consistent framework for stability. For time discrete training we show that if these properties fail to hold then methods may become unstable. Furthermore, using tools of $Γ$- convergence we provide new convergence results for weak solutions by only requiring that the neural network spaces are chosen to have suitable approximation properties. While our analysis is motivated by neural network-based approximation spaces, the framework developed here is applicable to any class of discrete functions satisfying the relevant approximation properties, and hence may serve as a foundation for the broader study of variational nonlinear PDE solvers.

math.NA↗

arXiv2025

Eigenvalue Inequalities for Fully Nonlinear Elliptic Equations via the Alexandroff-Bakelman-Pucci Method

In this work we establish eigenvalue inequalities for elliptic differential operators either for Dirichlet or for Robin eigenvalue problems, by using the technique introduced by Alexandroff, Bakelman and Pucci. These inequalities can be extended for fully nonlinear elliptic equations, such as for the Monge-Ampère equation and for Pucci's equations. As an application we establish a lower bound for the $ L^p -$norm of the Laplacian and this bound is sharp, in the sense that, when equality is achieved then a symmetry property is obtained. In addition, we obtain an $ L^{\infty} $ bound for the gradient of solutions to fully nonlinear elliptic equations and as a result, a $ C^3 $ estimate.

math.AP↗

arXiv2025

Applications of P-functions to Fully Nonlinear Elliptic equations: Gradient Estimates and Rigidity Results

We introduce the notion of $ P -$functions for fully nonlinear equations and establish a general criterion for obtaining such quantities for this class of equations. Some applications are gradient bounds, De Giorgi-type properties of entire solutions and rigidity results. In particular, we establish a pointwise gradient bound and a rigidity result for Pucci's equations. This pointwise gradient bound generalizes the Modica inequality in the case of fully nonlinear elliptic equations. Furthermore, we prove Harnack-type inequalities and local estimates for the gradient of solutions. In addition, we consider such quantities for higher order nonlinear equations and for equations of order greater than two we obtain Liouville-type theorems and pointwise estimates for the Laplacian.

math.AP↗

arXiv2025

Gradient Bounds and Liouville theorems for Quasi-linear equations on compact Manifolds with nonnegative Ricci curvature

In this work we establish a gradient bound and Liouville-type theorems for solutions to Quasi-linear elliptic equations on compact Riemannian Manifolds with nonnegative Ricci curvature. Also, we provide a local splitting theorem when the inequality in the gradient bound becomes equality at some point. Moreover, we prove a Harnack-type inequality and an ABP estimate for the gradient of solutions in domains contained in the manifold.

math.AP↗

arXiv2024

On the $ Γ$-convergence of the Allen-Cahn functional with boundary conditions

We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon $-energy functional with Dirichlet values and we establish the $ Γ$-limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves, we determine the precise structure of the minimizers of the limiting functional, and thus the limit of minimizers of the $ \varepsilon $-energy functional as $ \varepsilon \rightarrow 0 $.

math.AP↗

arXiv2023

A Relation of the Allen-Cahn equations and the Euler equations and applications of the Equipartition

We will prove that solutions of the Allen-Cahn equations that satisfy the equipartition can be transformed into solutions of the Euler equations with constant pressure. As a consequence, we obtain De Giorgi type results, that is, the level sets of entire solutions are hyperplanes. In addition, we obtain some examples of smooth entire solutions of the Euler equations in particular cases. For specific type of initial conditions, some of these solutions can be extended to the Navier-Stokes equations. Also, we will determine the structure of solutions of the Allen-Cahn system in two dimensions that satisfy the equipartition. Finally, we apply the Leray projection on the Allen-Cahn system and provide some explicit entire solutions.

math.AP↗

arXiv2021

Entire minimizers of Allen-Cahn systems with sub-quadratic potentials

We study entire minimizers of the Allen-Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal Euler-Lagrange equations are supplemented with free boundaries. We do not study regularity issues but focus on qualitative aspects. We show the existence of entire solutions in an equivariant setting connecting the minima of $ W $ at infinity, thus modeling many coexisting phases, possessing free boundaries and minimizing energy in the symmetry class. We also present a very modest result of existence of free boundaries under no symmetry hypotheses. The existence of a free boundary can be related to the existence of a specific sub-quadratic feature, a dead core, whose size is also quantified.

math.AP↗

arXiv2017

On Almost Entire Solutions Of The Burgers Equation

Solutions that satisfy classically the Burgers equation except, perhaps, on a closed set S of the plane of potential singularities whose Hausdorff 1-measure is zero, $H^1(S) = 0$, are necessarily identically constant. We show this under the additional hypothesis that $S$ is a subset of a countable union of ordered graphs.

math.AP↗