Search results for " equations"
showing 10 items of 783 documents
On time-harmonic Maxwell equations with nonhomogeneous conductivities : Solvability and FE-approximation
1989
The solvability of time-harmonic Maxwell equations in the 3D-case with nonhomogeneous conductivities is considered by adapting Sobolev space technique and variational formulation of the probJem in question. Moreover, a finite element approximation is presented in the 3D·case together with an error estimate in the energy norm. Some remarks are given to the 2D-problem arising from geophysics. peerReviewed
On a nonlinear Schrödinger equation for nucleons in one space dimension
2021
We study a 1D nonlinear Schrödinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, they are shown to appear in the long time behaviour of certain cases.
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
C1,α-regularity for variational problems in the Heisenberg group
2017
We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"older continuity of horizontal gradient of minima.
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).
On the regularity of very weak solutions for linear elliptic equations in divergence form
2020
AbstractIn this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .
Asymptotic mean value formulas for parabolic nonlinear equations
2021
In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge–Ampère equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of dynamic programming principles for certain two-player, zero-sum games. peerReviewed