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Register a new userMild solutions of semilinear elliptic equations in Hilbert spaces
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This paper extends the theory of regular solutions ($C^1$ in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of $G$-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into $G$-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now.
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In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,text{Dini}} domains.
In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n leq 9$. This result, that was only known to be true for $nleq4$, is optimal: $log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $ngeq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension $n leq 9$. This answers to two famous open problems posed by Brezis and Brezis-Vazquez.
We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that: - For $n=2$, there exist Morse index $1$ solutions whose $L^infty$ norm goes to infinity. - For $n geq 3$, uniform boundedness holds in the subcritical case for power-type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow-up in $L^infty$. In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori $L^infty$ bound for finite Morse index solution in the sharp dimensional range $3leq nleq 9$. As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method.
Given a smooth domain $OmegasubsetRR^N$ such that $0 in partialOmega$ and given a nonnegative smooth function $zeta$ on $partialOmega$, we study the behavior near 0 of positive solutions of $-Delta u=u^q$ in $Omega$ such that $u = zeta$ on $partialOmegasetminus{0}$. We prove that if $frac{N+1}{N-1} < q < frac{N+2}{N-2}$, then $u(x)leq C abs{x}^{-frac{2}{q-1}}$ and we compute the limit of $abs{x}^{frac{2}{q-1}} u(x)$ as $x to 0$. We also investigate the case $q= frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
We study a nonlinear equation in the half-space ${x_1>0}$ with a Hardy potential, specifically [-Delta u -frac{mu}{x_1^2}u+u^p=0quadtext{in}quad mathbb R^n_+,] where $p>1$ and $-infty<mu<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1to 0$, or is determined by the solutions of the linear problem $-Delta h -frac{mu}{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $mu$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like $O(x_1^{-2/(p-1)})$ near the origin and which, away from the origin have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like $O(x_1^{-2/(p-1)})$ at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.
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