In the past few decades, the uniqueness of positive solutions to elliptic equations with polynomial growth has been extensively studied. However, the corresponding problems associated with elliptic equations with critical exponential growth given by the Trudinger-Moser inequalities still remain open. For this kind of equations, the classic non-degeneracy method based on the Pohozaev identity and the study of the linearized equation do not seem to work. In this paper, we will solve this uniqueness problem. More precisely, we obtain uniqueness of positive solutions to equations of the form {-Delta u=lambda ue(u2 )x is an element of B-1 subset of R-2 u>0 x is an element of B-1 u=0 x is an element of partial derivative B-1 where 0
In this paper, we establish the radial symmetry of solutions to a free-boundary problem, characterized by a singular right-hand side, both in the elliptic and parabolic regimes. Specifically, we focus on the unit ball B-1 of R-n, where we examine a solution to the fully nonlinear elliptic problem given by: { F(D(2)u) = f(u) in B1 boolean AND {u > 0} u=M on partial derivative B-1 0 <= u in B-1. The right-hand side f (u) behaves near the free boundary partial derivative{u > 0} like u(a), with negative values for a is an element of (-1; 0). It is important to note that due to the lack of C-2-smoothness of both u and the free boundary partial derivative{u > 0}, the well-known Serrin-type boundary point lemma cannot be directly applied. To overcome this challenge, we devise an approach based on an exact assumption concerning the first-order expansion and the decay on the second order, complemented by an ad-hoc comparison principle. Furthermore, we extend our analysis to encompass the parabolic case of the problem, and present a corresponding result.
In this paper, we are concerned with a parabolic thin obstacle problem for the weighted biLaplace operator. Such problems are in connection with various physical applications such as the travelling waves phenomenon, pattern formation in spatially homogeneous systems, phase separation in isothermal binary systems, etc. We establish the well-posedeness of the problem, and we analyze the regularity of the solution depending on the dimension of the domain in consideration. Our methods are based on an implicit time-discretization approach
Let H be a separable Hilbert space. Building on the metric geometry of the Grassmannian P-c(H) of finite-dimensional subspaces of H, we develop a theory of optimal transport for the normal states of the von Neumann algebra of linear and bounded operators B(H). Seeing density matrices as discrete probability measures on P-c(H) (via the spectral theorem) we define an optimal transport cost and the Wasserstein distance for normal states. We prove that the transport cost induces the w*-topology and satisfies the triangular inequality on a dense subset of normal states. Our construction is compatible with the quantum mechanics approach of composite systems as tensor products. We provide a natural interpretation of the pure normal states of B(H circle times H) as families of transport maps. In this way we assign a Wasserstein cost to each pure normal state of B(H circle times H), consistently with the transport cost defined via P-c (H).
We describe explicitly the geometric compactifications, obtained by adding slc surfaces X with ample canonical class, for two connected components in the moduli space of surfaces of general type: Campedelli surfaces with pi(1)(X) = Z(2 )(3)and Burniat surfaces with K-2 = 6.
