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Karl K. Brustad

Publications and source records attributed to Karl K. Brustad.

13 recordsLinked to original sources
arXiv2026

Infinity-harmonic functions in the plane: Regularity by injectivity

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 that the gradient is locally $ α$-Hölder, but $ α$ comes without any positive lower bound. Aronsson's solution $ x^{4/3} - y^{4/3} $ shows that no better general regularity is possible. In the plane there is also a connection between the $ \infty $-Laplace equation and the one-dimensional heat equation, observed already by Aronsson himself. I shall show that this link can be accessed under a certain injectivity condition on the gradient, and that the caloric structure then is enough to prove the 1/3-Hölder continuity. Of course, an injective gradient is by no means a \emph{necessary} condition, as seen by smooth solutions such as the planes and cones.

math.AP↗
arXiv2023

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

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 the curved boundary. A recent explicit solution disproves a conjecture.

math.AP↗
arXiv2022

The Infinity-Potential in the Square

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 $u=1$ at the centre. This so-called $\infty$-potential is obtained with a hodograph method. The heat equation is used and one of Jacobi's Theta functions appears. The formula disproves a conjecture.

math.AP↗
arXiv2022

Counterexamples to the comparison principle in the special Lagrangian potential equation

For each $k = 0,\dots,n$ we construct a continuous phase $f_k$, with $f_k(0) = (n-2k)\fracπ{2}$, and viscosity sub- and supersolutions $v_k$, $u_k$, of the elliptic PDE $\sum_{i=1}^n \arctan(λ_i(D^2 w)) = f_k(x)$ such that $v_k-u_k$ has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases $f\colonΩ\to (-nπ/2,nπ/2)$. Our examples show it does not.

math.AP↗
arXiv2021

Sobolev gradients of viscosity supersolutions

We investigate which elliptic PDEs that have the property that every viscosity supersolution is $W^{1,q}_{loc}(Ω)$, $Ω\subseteq\mathbb{R}^n$. The asymptotic cone of the operator's sublevel set seems to be essential. It turns out that much can be said if we know how this cone compares to the sublevel set of a certain minimal operator associated with the exponent $q$.

math.AP↗
arXiv2019

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

The Dominative $p$-Laplacian is the operator defined for $2\le p < \infty$ as follows: \begin{equation}\label{dominativep} \mathcal{L}_{p}u(x)=\frac{1}{p}\left(λ_{1}+\ldots+λ_{N-1}\right)+\frac{(p-1)}{p}λ_{N}, \end{equation} where we have ordered the eigenvalues of $D^{2}u(x)$ as $λ_{1}\le λ_{2}\ldots\leλ_{N}$. The operator $\mathcal{L}_{p}u(x)$ was introduced by Brustand to give a natural explanation of the superposition principle for the $p$-Laplace equation. In this paper, we present a discrete stochastic approximation to the unique viscosity solution of the Dirichlet problem for the Dominative $p$-Laplace Equation.

math.AP↗
arXiv2019

Total derivatives of eigenvalues and eigenprojections of symmetric matrices

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 matrix-valued functions $H\colonΩ\to S(m)$ are given. The eigenvalues and eigenprojections are considered as functions in the same domain $Ω\subseteq\mathbb{R}^n$.

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arXiv2018

Sublinear Elliptic Operators

We investigate second order elliptic equations \[F(\mathcal{H}u) = 0\] where the function $F\colon S(n)\to\mathbb{R}$ on the space of symmetric $n\times n$ matrices is assumed to be sublinear. There is very little to be found in the literature devoted particularly to sublinear elliptic operators. When examples of such operators occur, they are often merely treated as members of the larger class of convex operators. That class has been thoroughly investigated and many of its aspects are well understood. There is, however, something to be said about sublinear operators that do not, in general, apply to convex operators.

math.AP↗
arXiv2017

Superposition of p-superharmonic functions

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 superposition principle in the $p$-Laplace Equation.

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arXiv2016

Superposition in the $p$-Laplace Equation

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. Zhang published their paper "Another Way to Say Harmonic" in 2003. We give a simple proof and extend the result by means of an explicit formula for the $p$-Laplacian of the superposition.

math.AP↗