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On compact manifolds (M,g) , we derive the existence of metrics in a given conformal class [g] with prescribed negative partial curvature. This curvature corresponds to a fully nonlinear equation derived from conformal geometry. For manifolds with boundary, we demonstrate the solvability of equations involving prescribed negative partial curvature within M , coupled with mean curvature along M .

In this paper, we prove the existence of weak solutions to Neumann boundary value problems for a nonlinear p(x)-elliptic equation of the form -div a(x,u,∇u)=b(x)| u |p(x)-2u+λH(x,u, ∇u). We established the existence result by using the topological degree introduced by Berkovits.

This paper explores a generalized nonlocal dispersion SIS epidemic model subject to the Neumann boundary conditions and spatial heterogeneity. We use a convolution operator to describe the nonlocal spatial movements of individuals. Our primary goal is to investigate this model, focusing on a generalized incidence function, which presents an additional challenge in the model analysis. This model’s basic reproduction number, R0, is identified, and it is proved that 1-R0 has the same sign as the principal eigenvalue of a generalized linear nonlocal operator. Furthermore, the asymptotic profiles of R0 in terms of dispersion coefficients are also established. We also investigate the existence and uniqueness of an endemic steady state for R0>1, and we study the large dispersal rates effect on the asymptotic profiles of the steady endemic state. Finally, we discussed the global asymptotic behavior of the solution for different dispersal coefficients.

In this paper, making use of the method developed by Catlin, we study the L 2-estimate for the ∂̅-equation on a lunar manifold with mixed boundary conditions. 2010 Mathematics Subject Classification. Primary 32W05; Secondary 32V15. Key words and phrases. ∂̅-operator, L 2-estimate, ∂̅-Dirichlet boundary condition, ∂̅-Neumann boundary condition.

We consider the initial-boundary value problem of the nonlinear Schrödinger equation on the half plane with a nonlinear Neumann boundary condition. After establishing the boundary Strichartz estimate in L 2 ( R + 2 ) and H s ( R + 2 ) , we consider the time local well-posedness of the problem in L 2 ( R + 2 ) and H s ( R + 2 ) .

We consider a Neumann problem for the fractional Laplacian involving a nonlocal Choquard-type nonlinearity and Sobolev–Hardy exponent. Under suitable assumptions on the data and using the Nehari manifold method, we discuss the existence problem in several subcritical and critical cases.

We construct the heat kernel on curvilinear polygonal domains in arbitrary surfaces for Dirichlet, Neumann, and Robin boundary conditions as well as mixed problems, including those of Zaremba type. We compute the short time asymptotic expansion of the heat trace and apply this expansion to demonstrate a collection of results showing that corners are spectral invariants. Nous construisons le noyau de la chaleur pour des domaines polygonaux curvilignes dans des surfaces arbitraires avec des conditions aux bords de Dirichlet, Neumann et Robin ainsi que des conditions mixtes, y compris celles de type Zaremba. Nous calculons l’expansion asymptotique de la trace quand le temps approche zéro et nous utilisons cette expansion pour démontrer un ensemble de résultats montrant que les coins sont des invariants spectraux.

We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace’s equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. Then, to regularize the integral, we multiply the Green’s function by a radial function with length parameter δ chosen so that the product is smooth. We show that a natural regularization has error O ( δ 3 ) , and a simple modification improves the error to O ( δ 5 ) . The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem, altered to account for the solvability condition, and evaluate the solution on the boundary. Choosing δ = c h 4 / 5 , we find about O ( h 4 ) convergence in our examples, where h is the spacing in a background grid.