Karl K. Brustad

activity

20162026

most citedSublinear Elliptic Operators

3 citations · 10 across the 13 of their papers we have counts for

collaborators

13 papers

math.AP2026

Infinity-harmonic functions in the plane: Regularity by injectivity

Karl K. Brustad

It has been a long standing conjecture that the $\infty$ -harmonic functions in the plane have a 1/3-Hölder continuous gradient. It \emph{is} known that solutions are $C^{1}$ and…

math.AP2023

The Infinity-Laplacian in Smooth Convex Domains and in a Square

Karl K. Brustad, Erik Lindgren, Peter Lindqvist

We extend some theorems for the Infinity-Ground State and for the Infinity-Potential, known for convex polygons, to other domains in the plane, by applying Alexandroff's method to…

math.AP2022

The Infinity-Potential in the Square

Karl K. Brustad

A representation formula for the solution of the $\infty$ -Laplace equation is constructed in a punctured square, the prescribed boundary values being $u = 0$ on the sides and $u = 1$ a…

math.AP2022

Counterexamples to the comparison principle in the special Lagrangian potential equation

Karl K. Brustad

For each $k = 0, \dots, n$ we construct a continuous phase $f_{k}$ , with $f_{k} (0) = (n - 2 k) \frac{π}{2}$ , and viscosity sub- and supersolutions $v_{k}$ , $u_{k}$ , of the elliptic PDE $\sum_{i=1}…

math.AP2021

Sobolev gradients of viscosity supersolutions

Karl K. Brustad

We investigate which elliptic PDEs that have the property that every viscosity supersolution is $W_{l oc}^{1, q} (Ω)$ , $Ω \subseteq R^{n}$ . The asymptotic cone of the operator's…

math.AP2020★ 1 cited

The one-dimensional nonlocal Dominative $p$ -Laplace equation

Karl K. Brustad

The explicit solution to the Dirichlet problem for a class of mean value equations on the real line is derived. It shed some light on the behavior of solutions to general nonlocal…

math.AP2020

On the comparison principle for second order elliptic equations without first and zeroth order terms

Karl K. Brustad

We consider the comparison principle for semicontinuous viscosity sub- and supersolutions of second order elliptic equations on the form $F (D^{2} w, x) = 0$ . A structural condition on…

math.AP2020

Segre's Theorem. An analytic proof of a result in differential geometry

Karl K. Brustad

We present an analytic approach on how to solve the problem $∣\nabla u ∣ = f (u)$ , $Δ u = g (u)$ , in connected domains $Ω \subseteq R^{n}$ .

math.AP2019★ 3 cited

Total derivatives of eigenvalues and eigenprojections of symmetric matrices

Karl K. Brustad

Conditions for existence and formulas for the first- and second order total derivatives of the eigenvalues, and the first order total derivatives of the eigenprojections of smooth…

math.AP2018★ 2 cited

A discrete stochastic interpretation of the Dominative $p$ -Laplacian

Karl K. Brustad, Peter Lindqvist, Juan J. Manfredi

The Dominative $p$ -Laplacian is the operator defined for $2 \leq p < \infty$ as follows: \begin{equation}\label{dominativep} \mathcal{L}_{p}u(x)=\frac{1}{p}\left(λ_{1}+\ldots+λ_{N-1}…

math.AP2018★ 3 cited

Sublinear Elliptic Operators

Karl K. Brustad

We investigate second order elliptic equations \[F(\mathcal{H}u) = 0\] where the function $F : S (n) \to R$ on the space of symmetric $n \times n$ matrices is assumed to b…

math.AP2017★ 1 cited

Superposition of p-superharmonic functions

Karl K. Brustad

The Dominative $p$ -Laplace Operator is introduced. This operator is a relative to the $p$ -Laplacian, but with the distinguishing property of being sublinear. It explains the superp…

math.AP2016

Superposition in the $p$ -Laplace Equation

Karl K. Brustad

That a superposition of fundamental solutions to the $p$ -Laplace Equation is $p$ -superharmonic -- even in the non-linear cases $p > 2$ -- has been known since M. Crandall and J. Zhan…

Infinity-harmonic functions in the plane: Regularity by injectivity

The Infinity-Laplacian in Smooth Convex Domains and in a Square

The Infinity-Potential in the Square

Counterexamples to the comparison principle in the special Lagrangian potential equation

Sobolev gradients of viscosity supersolutions

The one-dimensional nonlocal Dominative p-Laplace equation

On the comparison principle for second order elliptic equations without first and zeroth order terms

Segre's Theorem. An analytic proof of a result in differential geometry

Total derivatives of eigenvalues and eigenprojections of symmetric matrices

A discrete stochastic interpretation of the Dominative p-Laplacian

Sublinear Elliptic Operators

Superposition of p-superharmonic functions

Superposition in the p-Laplace Equation

The one-dimensional nonlocal Dominative $p$ -Laplace equation

A discrete stochastic interpretation of the Dominative $p$ -Laplacian

Superposition in the $p$ -Laplace Equation