Narrow Results

Let $B_n$ be the open unit ball in ${ℂ}^n$ and let $(M^m,g)$ be a Kähler manifold with strongly negative or strongly semi-negative curvature. In this paper, we study Siu type rigidity theorem for the harmonic map $u\in C^2(\overline{B_n},M)$ satisfying the boundary condition that ${\sum }_{i,j=1}^n{z}_i{\bar{z}}_j{u}_{i\bar{j}}=0$ on $\partial B_n$. We also prove the existence and uniqueness theorem for some Neumann type boundary value problem for harmonic functions on $B_n$.

We study the Neumann problem for complex special Lagrangian equations with critical phase. A priori derivative estimates up to order two are established. And we obtain the existence and uniqueness theorem by the method of continuity.

There are three over-determined boundary value problems for the inhomogeneous Cauchy–Riemann equation, the Dirichlet problem, the Neumann problem and the Robin problem. These problems have found their significant importance and applications in diverse fields of science, for example, applied mathematics, physics, engineering and medicine. In this paper, we explicitly investigate the solvability of these three basic boundary value problems for the Cauchy–Riemann operator on an isosceles orthogonal triangle. The solvability conditions are explicitly obtained. The plane parqueting reflection principle is used. Cauchy–Pompeiu-type formulae for the triangle are established. The boundary behavior of a related integral operator of the Schwarz type is studied in detail. Then, the boundary values of solutions at the corner points are found.

This paper concerns the gradient estimates for Neumann problem of a certain Monge–Ampère type equation with a lower order symmetric matrix function in the determinant. Under a one-sided quadratic structure condition on the matrix function, we present two alternative full discussions of the global gradient bound for the elliptic solutions.

Let $n\ge 3$, $\mathrm{\Omega }$ be a bounded (semi-)convex domain in ${ℝ}^n$ and the non-negative potential $V$ belong to the reverse Hölder class ${\mathrm{\text{RH}}}_n({ℝ}^n)$. Assume that $p\in (1,\infty )$ and $\omega \in A_p(\partial \mathrm{\Omega })$, where $A_p(\partial \mathrm{\Omega })$ denotes the Muckenhoupt weight class on $\partial \mathrm{\Omega }$, the boundary of $\mathrm{\Omega }$. In this paper, the authors show that, for any $p\in (1,\infty )$, the Neumann problem for the Schrödinger equation $-\mathrm{\Delta }u+Vu=0$ in $\mathrm{\Omega }$ with boundary data in (weighted) $L^p$ is uniquely solvable. The obtained results in this paper essentially improve the known results which are special cases of the results obtained by Shen [Indiana Univ. Math. J.43 (1994) 143–176] and Tao and Wang [Canad. J. Math.56 (2004) 655–672], via extending the range $p\in (1,2]$ of $p$ into $p\in (1,\infty )$.

In this paper, we prove the existence of the $C^{1,1}$-solution to the classical Neumann problem for the degenerate elliptic Hessian quotient equation $\frac{{\sigma }_k(D^{2}u)}{{\sigma }_l(D^{2}u)}=f(x)$ under the condition that $f^{\frac{1}{k-l}}\in C^2(\overline{\mathrm{\Omega }})$.

We investigate the existence of positive solutions for a class of Minkowski-curvature equations with indefinite weight and nonlinear term having superlinear growth at zero and super-exponential growth at infinity. As an example, for the equation

In this note we obtain a multiplicity result for an eigenvalue Neumann problem. Precisely, a recent critical point result for differentiable functionals is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three non-zero weak solutions.

In this paper, by applying the well-known method for dealing with $p$-Laplace type elliptic boundary value problems, the authors establish a sharp estimate for the decreasing rearrangement of the gradient of solutions to the Dirichlet and the Neumann boundary value problems of a class of Schrödinger equations, under the weak regularity assumption on the boundary of domains. As applications, the gradient estimates of these solutions in Lebesgue spaces and Lorentz spaces are obtained.

This paper is concerned wih the global existence and the partial regularity for the weak solution of the Landau-Lifshitz-Maxell system in two dimensions with Neumann boundary conditions.

In this paper, we study a class of generalized and not necessarily differentiable functionals of the form

In this paper, we obtain some important inequalities of Hessian quotient operators, and global $C^2$ estimates of the Neumann problem of Hessian quotient equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problem of Hessian quotient equations.

The aim of this paper is to present some multiplicity results for elliptic boundary value problems involving oscillating nonlinearities. These results are obtained by using a variational theorem of Ricceri and some developments of this latter.

Let Ω ⊂ ℝ^N be a bounded open set. We deal with the existence of weak solutions for the following Neumann problem

The aim of this paper is to provide suitable conditions under which the following Neumann problem:

Using the harmonic and biharmonic Neumann functions N₁(z, ζ) and N₂(z, ζ), and operators having these functions as kernels, the Neumann problem for second and fourth order complex partial differential equations with main parts as the harmonic and biharmonic operators, are discussed. Solvability conditions are obtained.

Let B be a ball in a two dimensional hemisphere or in a two dimensional hyperbolic space. Let f ∈ C². We prove the following nonlinear “hot spots” conjecture on B: If the nonconstant solution u of the Neumann problem