ﻻ يوجد ملخص باللغة العربية
We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) = psi(x)$ where $f$ is a natural differential operator, with a restricted domain $F$, on a manifold $X$. By natural we mean operators that arise intrinsically from a given geometry on $X$. An important point is that the equation need not be convex and can be highly degenerate. Furthermore, the inhomogeneous term can take values at the boundary of the restricted domain $F$ of the operator $f$. A simple example is the real Monge-Amp`ere operator ${rm det}({rm Hess},u) = psi(x)$ on a riemannian manifold $X$, where ${rm Hess}$ is the riemannian Hessian, the restricted domain is $F = {{rm Hess} geq 0}$, and $psi$ is continuous with $psigeq0$. A main new tool is the idea of local jet-equivalence, which gives rise to local weak comparison, and then to comparison under a natural and necessary global assumption. The main theorem applies to pairs $(F,f)$, which are locally jet-equivalent to a given constant coefficient pair $({bf F}, {bf f})$. This covers a large family of geometric equations on manifolds: orthogonally invariant operators on a riemannian manifold, G-invariant operators on manifolds with G-structure, operators on almost complex manifolds, and operators, such as the Lagrangian Monge-Amp`ere operator, on symplectic manifolds. It also applies to all branches of these operators. Complete existence and uniqueness results are established with existence requiring the same boundary assumptions as in the homogeneous case [10]. We also have results where the inhomogeneous term $psi$ is a delta function.
In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $mathbb{R}^n$.
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particula
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $Mtimes mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solu
We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the
We make the split of the integral fractional Laplacian as $(-Delta)^s u=(-Delta)(-Delta)^{s-1}u$, where $sin(0,frac{1}{2})cup(frac{1}{2},1)$. Based on this splitting, we respectively discretize the one- and two-dimensional integral fractional Laplaci