Search results for "partial differential equations"

Systematic derivation of partial differential equations for second order boundary value problems

2022

Software systems designed to solve second order boundary value problems are typically restricted to hardwired lists of partial differential equations. In order to come up with more flexible systems, we introduce a systematic approach to find partial differential equations that result in eligible boundary value problems. This enables one to construct and combine one's own partial differential equations instead of choosing those from a pre-given list. This expands significantly end users possibilities to employ boundary value problems in modeling. To introduce the main ideas we employ differential geometry to examine the mathematical structure involved in second order boundary value problems …

From Particle Systems to Partial Differential Equations International Conference, Particle Systems and PDEs VI, VII and VIII, 2017-2019

2021

This book includes the joint proceedings of the International Conference on Particle Systems and PDEs VI, VII and VIII. Particle Systems and PDEs VI was held in Nice, France, in November/December 2017, Particle Systems and PDEs VII was held in Palermo, Italy, in November 2018, and Particle Systems and PDEs VIII was held in Lisbon, Portugal, in December 2019. Most of the papers are dealing with mathematical problems motivated by different applications in physics, engineering, economics, chemistry and biology. They illustrate methods and topics in the study of particle systems and PDEs and their relation. The book is recommended to probabilists, analysts and to those mathematicians in general…

Equivalence of viscosity and weak solutions for the normalized $p(x)$-Laplacian

2018

We show that viscosity solutions to the normalized $p(x)$-Laplace equation coincide with distributional weak solutions to the strong $p(x)$-Laplace equation when $p$ is Lipschitz and $\inf p>1$. This yields $C^{1,\alpha}$ regularity for the viscosity solutions of the normalized $p(x)$-Laplace equation. As an additional application, we prove a Rad\'o-type removability theorem.

On the local and global regularity of tug-of-war games

2018

This thesis studies local and global regularity properties of a stochastic two-player zero-sum game called tug-of-war. In particular, we study value functions of the game locally as well as globally, that is, close to the boundaries of the game domains. Furthermore, we formulate a continuous time stochastic differential game and discuss, among other things, the equicontinuity of the families of value functions. The main motivation is to understand the properties of the games on their own right. As applications, we obtain an existence and a regularity result for a nonlinear elliptic p-Laplace type partial differential equation and a characterization of the solution to a parabolic p-Laplace typ…

Gradient walks and $p$-harmonic functions

2017

Nonlinear Liouville Problems in a Quarter Plane

2016

We answer affirmatively the open problem proposed by Cabr\'e and Tan in their paper "Positive solutions of nonlinear problems involving the square root of the Laplacian" (see Adv. Math. {\bf 224} (2010), no. 5, 2052-2093).

Shape optimization utilizing consistent sensitivities

2010

A graph-based multigrid with applications

2010

Remarks on regularity for p-Laplacian type equations in non-divergence form

2018

We study a singular or degenerate equation in non-divergence form modeled by the $p$-Laplacian, $$-|Du|^\gamma\left(\Delta u+(p-2)\Delta_\infty^N u\right)=f\ \ \ \ \text{in}\ \ \ \Omega.$$ We investigate local $C^{1,\alpha}$ regularity of viscosity solutions in the full range $\gamma>-1$ and $p>1$, and provide local $W^{2,2}$ estimates in the restricted cases where $p$ is close to 2 and $\gamma$ is close to 0.