arXiv research

Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.

Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.

The paper proves Liouville theorems and nonexistence results for semilinear equations on pseudo-Hermitian manifolds.

Paper proves no nontrivial solutions to certain elliptic equations on graphs.

The paper classifies solutions to a specific elliptic equation in the Heisenberg group.

The paper classifies solutions to semilinear equations on curved spaces.

Deep neural nets solve complex insurance math equations.

We consider the Dirichlet problem for semilinear elliptic equations on a bounded domain which is diffeomorphic to a ball and investigate bifurcation from a given (trivial) branch of solutions, where the radius of the ball serves as bifurcation parameter. Our methods are based on well known results from variational bifu…

Study on ground states of semilinear elliptic equations with various potential wells.

Study solves inverse problems for equations with fractional nonlinearities.

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. Th…

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an $n$-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth $(n-2)$-dimensional subm…

New Kelvin transform for anisotropic elliptic problems.

We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not…

Assume that $f(s) = F'(s)$ where $F$ is a double-well potential. Under certain conditions on the Lipschitz constant of $f$ on $[-1,1]$, we prove that arbitrary bounded global solutions of the semilinear equation $Δu = f(u)$ on hyperbolic space $\HH^n$ must reduce to functions of one variable provided they admit asympto…

We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.

Develops comparison methods for semilinear elliptic problems on Riemannian manifolds with Ricci lower bound.

The article derives gradient estimations for semilinear equations on geometric flows.

Derives matrix Harnack inequalities for semilinear heat equations on manifolds.

Trivial solution proof for heat equation on certain manifolds.

Derives Li & Yau estimates for heat equations on manifolds.

Sharp conditions found for solving heat equation on Riemannian manifolds.

Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.

Study well-posedness of fast diffusion equation on noncompact manifolds.

Classifies self-similar solutions for heat equations with positive speed.

On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is {\it not} in divergence form. The case of noncompact complete preimage and target manifolds is considered. We give conditio…

Improved convergence and curvature estimate for parabolic Allen-Cahn equation.

Study semilinear equations on weighted manifolds to prove rigidity.

Measuring wave sources uniquely identifies manifold properties.

In this paper, we investigate the problem of blow up and sharp upper bound estimates of the lifespan for the solutions to the semilinear wave equations, posed on asymptotically Euclidean manifolds. Here the metric is assumed to be exponential perturbation of the spherical symmetric, long range asymptotically Euclidean …

We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian manifolds. As a byproduct of the proofs we also obtain some rigidity, or partial symmetry…

We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of vari…

New approach simplifies proof of wave equations on black holes.

In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold $M^{2m}$. Such objects satisfy the elliptic system weakly $[J, Δ^m J]=0$. We prove a very general regularity theorem for semilinear systems in critical dimensions (with \emph{cr…

The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectra…

Paper introduces a new method to solve complex PDEs efficiently.

The paper establishes scattering theory for wave equations on Schwarzschild spacetime.

The aim of this paper is to prove the existence of weak solutions to the equation $Δu + u^p = 0$ which are positive in a domain $Ω\subset {\Bbb R}^N$, vanish at the boundary, and have prescribed isolated singularities. The exponent $p$ is required to lie in the interval $(N/(N-2), (N+2)/(N-2))$. We also prove the exist…

Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs

Study asymptotically almost periodic solutions on real hyperbolic manifolds.

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.

The goal of this paper is to clarify when a semilinear stochastic partial differential equation driven by Lévy processes admits an affine realization. Our results are accompanied by several examples arising in natural sciences and economics.

In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the $γ_k$ functions on the space of its hermitian metrics.

We extend the viscosity solution characterization proved in [5] for call/put American option prices to the case of a general payoff function in a multi-dimensional setting: the price satisfies a semilinear re-action/diffusion type equation. Based on this, we propose two new numerical schemes inspired by the branching p…

Paper solves portfolio problem using improved stochastic methods.

Let $(M,g)$ be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on $\del M$ and Weingarten foliations in some neighbourhood of infinity in $M$. We focus mostly on foliations where each lea…

Study phase transitions with prescribed mean curvature in Riemannian manifolds.

New method solves complex curvature equations.