Uniqueness of sign-changing solutions to Trudinger's equation
We establish uniqueness for sign-changing solutions to Trudinger's parabolic equation with time dependent $C^2$ Dirichlet boundary data.
math.AP↗
arXiv subjects
Peter Lindqvist
Publications and source records attributed to Peter Lindqvist.
Trudinger's Parabolic Equation
We study the uniqueness of non-negative solutions of the equation \begin{align*} \partial_t\left(|u|^{p-2}u\right)\,=\, \operatorname{div}(|\nabla u|^{p-2}\nabla u). \end{align*} Basic estimates are derived with the Galerkin Method.
math.AP↗
Regularity of Supersolutions
The regularity for the supersolutions of the Evolutionary p-Laplace Equation is considered. In particular,the equivalence of viscosity supersolutions and p-supercaloric functions (lower semicontinuous supersolutions defined via a comparison principle) is considered. Bounded viscosity supersolutions are, in fact, weak supersolutions belonging to a natural Sobolev Space.
math.AP↗
Lipschitz continuity and equivalence of positive viscosity and weak solutions to Trudinger's equation
We show that stricty positive viscosity supersolutions to Trudinger's equation are local weak supersolutions when $1<p<\infty$. As an application, we show using the Ishii-Lions method that not only viscosity but also weak solutions are Lipschitz continuous in the space variable. To the best of our knowledge, there is no preceding valid Lipschitz proofs in the literature.
math.AP↗
On the uniqueness of eigenfunctions for the vectorial $p$-Laplacian
We study a non-linear eigenvalue problem for vector-valued eigenfunctions and give a succinct uniqueness proof for minimizers of the associated Rayleigh quotient.
math.AP↗
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↗
Two formulas for plane paritions
We give a direct deduction and proof of two identities in the theory of plane partitions. The first one is known to enumerate the traces of plane partitions. The second one comes without any combinatorial interpretation.
math.CO↗
On $\infty$-Ground States in the Plane
We study $\infty$-Ground states in convex domains in the plane. In a polygon, the points where an $\infty$-Ground state does not satisfy the $\infty$-Laplace Equation are characterized: they are restricted to lie on specific curves, which are acting as attracting (fictitious) streamlines. The gradient is continuous outside these curves and no streamlines can meet there.
math.AP↗
The Gradient Flow of Infinity-Harmonic Potentials
We study the streamlines of $\infty$-harmonic functions in planar convex rings. We include convex polygons. The points where streamlines can meet are characterized: they lie on certain curves. The gradient has constant norm along streamlines outside the set of meeting points, the infinity-ridge.
math.AP↗
On a comparison principle for Trudinger's equation
We study the comparison principle for non-negative solutions of the equation $$ \frac{\partial\,(|v|^{p-2}v)}{\partial t}\,=\, \textrm{div} (|\nabla v|^{p-2}\nabla v), \quad 1<p<\infty.$$ This equation is related to extremals of Poincaré inequalities in Sobolev spaces. We apply our result to obtain pointwise control of the large time behavior of solutions.
math.AP↗
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↗
A test against trend in random sequences
We study a modification of Kendall's tau-test, replacing his permutations of n different numbers by sequences of length n, where repetition is allowed. In particular, binary sequences are included. Random sequences can be tested.
math.ST↗
Infinity-Harmonic Potentials and Their Streamlines
We consider certain solutions of the Infinity-Laplace Equation in planar convex rings. Their ascending streamlines are unique while the descending ones may bifurcate. We prove that bifurcation occurs in the generic situation and as a consequence, the solutions cannot have Lipschitz continuous gradients.
math.AP↗
A remark on infinite initial values for quasilinear parabolic equations
We study the possibility of prescribing infinite initial values for solutions of the Evolutionary $p$-Laplace Equation in the fast diffusion case $p>2$. This expository note has been extracted from our previous work. When infinite values are prescribed on the whole initial surface, such solutions can exist only if the domain is a space-time cylinder.
math.AP↗
Regularity of Solutions of the Parabolic Normalized p-Laplace Equation
The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to $L^2$.
math.AP↗
Regularity for an anisotropic equation in the plane
We present a simple proof of the $C^1$ regularity of $p$-anisotropic functions in the plane for $2\leq p<\infty$. We achieve a logarithmic modulus of continuity for the derivatives. The monotonicity (in the sense of Lebesgue) of the derivatives is used. The case with two exponents is also included.
math.AP↗
Supercaloric functions for the porous medium equation
In the slow diffusion case unbounded supersolutions of the porous medium equation are of two totally different types, depending on whether the pressure is locally integrable or not. This criterion and its consequences are discussed.
math.AP↗
The Time Derivative in a Singular Parabolic Equation
We study the Evolutionary p-Laplace Equation in the singular case 1 < p < 2. We prove that a weak solution has a time derivative in Sobolev's sense and that the time derivative is locally summable to some power > 1.
math.AP↗