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Register a new userNonexistence results for elliptic differential inequalities with a potential in bounded domains
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In this paper we are concerned with a class of elliptic differential inequalities with a potential in bounded domains both of $mathbf{R}^m$ and of Riemannian manifolds. In particular, we investigate the effect of the behavior of the potential at the boundary of the domain on nonexistence of nonnegative solutions.
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We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $Omegatimes(0,+infty)$, where $Omega$ is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.
We derive a diffusion approximation for the kinetic Vlasov-Fokker-Planck equation in bounded spatial domains with specular reflection type boundary conditions. The method of proof involves the construction of a particular class of test functions to be chosen in the weak formulation of the kinetic model. This involves the analysis of the underlying Hamiltonian dynamics of the kinetic equation coupled with the reflection laws at the boundary. This approach only demands the solution family to be weakly compact in some weighted Hilbert space rather than the much tricky $mathrm L^1$ setting.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
In this paper we study a sharp Hardy-Littlewood-Sobolev (HLS) type inequality with Riesz potential on bounded smooth domains. We obtain the inequality for a general bounded domain $Omega$ and show that if the extension constant for $Omega$ is strictly larger than the extension constant for the unit ball $B_1$ then extremal functions exist. Using suitable test functions we show that this criterion is satisfied by an annular domain whose hole is sufficiently small. The construction of the test functions is not based on any positive mass type theorems, neither on the nonflatness of the boundary. By using a similar choice of test functions with the Poisson-kernel-based extension operator we prove the existence of an abstract domain having zero scalar curvature and strictly larger isoperimetric constant than that of the Euclidean ball.
Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(Sigma,g)$. Precisely, if $lambda_1(Sigma)$ is the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition, then there exists a positive real number $alpha^ast<lambda_1(Sigma)$ such that for all $alphain (alpha^ast,lambda_1(Sigma))$, the supremum $$sup_{uin W^{1,2}(Sigma,g),,int_Sigma udv_g=0,,| abla_gu|_2leq 1}int_Sigma exp(4pi u^2(1+alpha|u|_2^2))dv_g$$ can not be attained by any $uin W^{1,2}(Sigma,g)$ with $int_Sigma udv_g=0$ and $| abla_gu|_2leq 1$, where $W^{1,2}(Sigma,g)$ denotes the usual Sobolev space and $|cdot|_2=(int_Sigma|cdot|^2dv_g)^{1/2}$ denotes the $L^2(Sigma,g)$-norm. This complements our earlier result in [39].
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