Related papers: The Dirichlet problem for mixed Hessian equations …

In [18], fundamental solutions for the generalized bi-axially symmetric Helmholtz equation were constructed in $R_2^ + = \left\{ {\left( {x,y} \right):x > 0,y > 0} \right\}.$ They contain Kummer's confluent hypergeometric functions in three…

In this note we provide uniform a priori estimates for solutions to degenerate complex Hessian equations on compact hermitian manifolds. Our approach relies on the corresponding a priori estimates for Monge-Amp\`ere equations; it provides…

Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of elliptic equations on compact hyperhermitian manifolds. By adapting the approach of Sz\'ekelyhidi to the hypercomplex setting, we prove some a priori estimates…

The main result of this note is the existence of martingale solutions to the stochastic heat equation (SHE) in a Riemannian manifold by using suitable Dirichlet forms on the corresponding path/loop space. Moreover, we present some…

We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance…

The article considers the Dirichlet problem for a high-order mixed-type equation that splits into factors, each of which is a Lavrentiev-Bitsadze equation with its own excellent coefficient. Sufficient conditions are found for the…

A priori estimates for the weak solutions the Dirichlet problem for the uniformly higher-order elliptic equations in a smooth bounded domain $\Omega\subset \Rn$ in generalized weighted Sobolev-Morrey spaces are obtained.

A viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in $\C^n$. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by…

We establish existence and multiplicity of solutions for a elliptic resonant elliptic problem under Dirichlet boundary conditions.

Hermitian symmetric manifolds are Hermitian manifolds which are homogeneous and such that every point has a symmetry preserving the Hermitian structure. The aim of these notes is to present an introduction to this important class of…

Let $X$ be a smooth $n\,$-dimensional manifold and $D$ be an open connected set in $X$ with smooth boundary $\partial D$. Perturbing the Cauchy problem for an elliptic system $Au = f$ in $D$ with data on a closed set $\iG \subset \partial…

In this paper, we investigate the continuity of solutions to the Dirichlet problem for complex Hessian-type equations associated with $(\omega, m)-\beta$-subharmonic functions on a ball in $\mathbb{C}^n$, where $ \beta=d…

The Hermite-Birkhoff interpolation problem of a function given on arbitrarily distributed points on the sphere and other manifolds is considered. Each proposed interpolant is expressed as a linear combination of basis functions, the…

In this paper, we consider the Dirichlet problem for a class of prescribed curvature equations. Both degenerate and non-degenerate cases are considered. The existence of the $C^{1,1}$ regular graphic hypersurfaces with prescribing a class…

In this paper, we establish global C^2 estimates for a class of mixed Hessian equations with Neumann boundary condition, and obtain the existence theorem of k-admissible solutions for the classical Neumann problem of these mixed Hessian…

We consider the Dirichlet problem in an ellipsoidal cylinder when the data function is entire. Under an additional assumption that the order of the data function is less than one, we show that there is a solution that extends as an entire…

In this paper, we establish the modified concavity inequality for complex Hessian equations under the semi-convexity assumption inspired by Lu \cite{Lu23} and Zhang \cite{Z24} for real case. Then second order estimates for admissible…

In this paper, we introduce $m$-subharmonic functions in quaternionic space $\mathbb{H}^{n}$, we define the quaternionic Hessian operator and solve the homogeneous Dirichlet problem for the quaternionic Hessian equation on the unit ball…

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of…

We solve the Dirichlet problem in the unit disc and derive the Poisson formula using very elementary methods and explore consequent simplifications in other foundational areas of complex analysis.