arXiv research

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

New Kelvin transform for anisotropic elliptic problems.

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

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.

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…

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

Deep neural nets solve complex insurance math equations.

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.

Study solves inverse problems for equations with fractional nonlinearities.

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…

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…

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

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…

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…

Study phase transitions with prescribed mean curvature in Riemannian manifolds.

We continue our previous work studying critical exponent semilinear elliptic (and subelliptic) problems which generalize the classical Yamabe problem. In [3] the focus was on metric-measure spaces with an `almost smooth' structure, with stratified spaces furnishing the key examples. The criterion for solvability there …

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 paper explores symmetry in solutions of semilinear PDEs on Riemannian domains.

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…

Sharp conditions found for solving heat equation on Riemannian manifolds.

NO approximates non-Markovian BSDEs with polynomial scaling in 1/ε.

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

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…

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

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…

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

Derives matrix Harnack inequalities for semilinear heat equations on manifolds.

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 …

Derives Li & Yau estimates for heat equations on manifolds.

New approach simplifies proof of wave equations on black holes.

Trivial solution proof for heat equation on certain manifolds.

Study semilinear equations on weighted manifolds to prove rigidity.

On an affine flat manifold with coordinates x^j and convex local potential function f, we call the affine Kahler metric f_{ij} dx^i dx^j semi-flat Calabi-Yau if it satisfies det f_{ij} = 1. Recently Gross-Wilson have constructed many such metrics on S^2 minus 24 singularities, as degenerate limits of Calabi-Yau metrics…

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…

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…

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

Measuring wave sources uniquely identifies manifold properties.

Nonexistence results for semilinear parabolic and hyperbolic inequalities on metric graphs

We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted …

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

Paper solves portfolio problem using improved stochastic methods.

We consider radial solutions to the fast diffusion equation $u_t=Δu^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance $r$ from a given point $o \in \mathbb{H}^N$. We investigate their fine asymptotics near…

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…

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

In this paper, we construct a Rabinowitz-Floer type homology for a class of non-linear problems having a \emph{starshaped} potential; we consider some equivariant cases as well. We give an explicit computation of the homology and we apply it to obtain results of existence and multiplicity of solutions for several model…