Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLiouville theorems on the upper half space
237
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we shall establish some Liouville theorems for solutions bounded from below to certain linear elliptic equations on the upper half space. In particular, we show that for $a in (0, 1)$ constants are the only $C^1$ up to the boundary positive solutions to $div(x_n^a abla u)=0$ on the upper half space.
rate research
Read More
In this paper we shall classify all positive solutions of $ Delta u =a u^p$ on the upper half space $ H =Bbb{R}_+^n$ with nonlinear boundary condition $ {partial u}/{partial t}= - b u^q $ on $partial H$ for both positive parameters $a, b>0$. We will prove that for $p ge {(n+2)}/{(n-2)}, 1leq q<{n}/{(n-2)}$ (and $n ge 3$) all positive solutions are functions of last variable; for $p= {(n+2)}/{(n-2)}, q= {n}/{(n-2)}$ (and $n ge 3$) positive solutions must be either some functions depending only on last variable, or radially symmetric functions.
There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $lambda=n-alpha$ (that is for the case of $alpha>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.
We study a class of second-order degenerate linear parabolic equations in divergence form in $(-infty, T) times mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-infty, T) times partial mathbb R^d_+$, where $mathbb R^d_+ = {x in mathbb R^d,:, x_d>0}$ and $Tin {(-infty, infty]}$ is given. The coefficient matrices of the equations are the product of $mu(x_d)$ and bounded uniformly elliptic matrices, where $mu(x_d)$ behaves like $x_d^alpha$ for some given $alpha in (0,2)$, which are degenerate on the boundary ${x_d=0}$ of the domain. Under a partially VMO assumption on the coefficients, we obtain the wellposedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.
In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and Yan in cite{HWY2008}) by introducing a uniform approach. The extremal functions are classified via the method of moving spheres, and the best constants are computed. The new approach can also be applied to obtain the classical HLS inequality and other similar inequalities.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
