Existence and Regularity Results for p-Kirchhoff Problems with Singular Nonlinearity

Let and consider the following anisotropic p-Laplace equation Under suitable hypothesis on the weight function g we present an existence result for in a bounded smooth domain Ω and nonexistence results for or with respectively.

In this paper, we delve into a discrete nonlinear singular semipositone problem, characterized by a nonlinear boundary condition. The nonlinearity, given by f(u)−auα with α>0, exhibits a singularity at u=0 and tends towards −∞ as u approaches 0+. By constructing some suitable auxiliary problems, the difficulty that arises from the singularity and semipositone of nonlinearity and the lack of a maximum principle is overcome. Subsequently, employing the Krasnosel’skii fixed-point theorem, we determine the parameter range that ensures the existence of at least one positive solution and the emergence of at least two positive solutions. Furthermore, based on our existence results, one can obtain the symmetry of the solutions after adding some symmetric conditions on the given functions by using a standard argument.

We examine the convergence of collocation with polynomials for elliptic problems. The general theory is imbedded in theL p -theory from which known results of existence and regularity are adopted. Convergence results are deduced in the cases of periodic, Dirichlet and mixed boundary conditions. It becomes obvious that the order of convergence essentially depends on the smoothness of the solution.

Our approach to boundary problems of finite elastostatics is of local type; hence, it requires the existence of deformations at which the linearized operator is a homeomorphism between suitable Banach spaces. The aim of this chapter is to provide appropriate results of existence, uniqueness, and regularity for the principal boundary problems of linearized elastostatics: namely, for the Dirichlet and Neumann problems.

We study time-dependent flows of incompressible degenerate power-law fluids characterized by the power-law index $p-2$ with $p>2$. In this case, the generalized viscosity vanishes as (the modulus of) the shear rate tends to zero. We prove global-in-time existence of a weak solution if $p>\max\{\frac{3d-4}{d},2\}$. This improves the range $p>\frac{3d+2}{d+2}$ for which the existence result was obtained by O.A.\ Ladyzhenskaya and J.L.\ Lions, via standard monotone operator theory. Since we apply higher differentiability techniques, certain regularity results are also established. The key step of the proof is an estimate of the velocity gradient in a suitable Nikol$'$skĭ space. To make the presentation of the method transparent, we restrict ourselves to the spatially periodic problem. A possible extension of the approach to no-slip boundary conditions is however discussed as well.

The precipitating quasi-geostrophic equations go beyond the (dry) quasi-geostrophic equations by incorporating the effects of moisture. This means that both precipitation and phase changes between a water-vapour phase (outside a cloud) and a water-vapour-plus-liquid phase (inside a cloud) are taken into account. In the dry case, provided that a Laplace equation is inverted, the quasi-geostrophic equations may be formulated as a nonlocal transport equation for a single scalar variable (the potential vorticity). In the case of the precipitating quasi-geostrophic equations, inverting the Laplacian is replaced by a more challenging adversary known as potential-vorticity-andmoisture inversion. The PDE to invert is nonlinear and piecewise elliptic with jumps in its coefficients across the cloud edge. However, its global ellipticity is a priori unclear due to the dependence of the phase boundary on the unknown itself. This is a free boundary problem where the location of the cloud edge is one of the unknowns. Here we present the first rigorous analysis of this PDE, obtaining existence, uniqueness, and regularity results. In particular the regularity results are nearly sharp. This analysis rests on the discovery of a variational formulation of the inversion. This novel formulation is used to answer a key question for applications: which quantities jump across the interface and which quantities remain continuous? Most notably we show that the gradient of the unknown pressure, or equivalently the streamfunction, is Hölder continuous across the cloud edge.

For partitions of space into sets of finite perimeter, and for general convex surface energy density functions, we show that surface energy may be locally linearly approximated, with uniform error bounds which hold even when the interfaces and surface energy density functions are non-smooth. We show that locally simple partitions – those satisfying a quite general density ratio condition – have boundaries which locally almost everywhere consist of a single interface, and have boundary closures which are (n − 1) dimensional. Local simplicity typically holds for variational problems in which surface area, surface energy or a surface plus bulk energy sum are to be minimized. Local simplicity rules out pathological boundary behaviour, helps establish lower semicontinuity, existence and regularity results and facilitates the use of finite perimeter partitions in applications in materials science, image processing and other fields.

In this paper, we study two Energy Balance Models with Memory arising in climate dynamics, which consist in a 1D degenerate nonlinear parabolic equation involving a memory term, and possibly a set-valued reaction term (of Sellers type and of Budyko type, in the usual terminology). We provide existence and regularity results, and obtain uniqueness and stability estimates that are useful for the determination of the insolation function in Sellers’ model with memory.

We consider finite horizon stochastic mean field games in which the state space is a network. They are described by a system coupling a backward in time Hamilton–Jacobi–Bellman equation and a forward in time Fokker–Planck equation. The value function u is continuous and satisfies general Kirchhoff conditions at the vertices. The density m of the distribution of states satisfies dual transmission conditions: in particular, m is generally discontinuous across the vertices, and the values of m on each side of the vertices satisfy some compatibility conditions. The stress is put on the case when the Hamiltonian is Lipschitz continuous. Existence, uniqueness and regularity results are proven.

This paper is devoted to a scalar model of the Oseen equations, a linearized form of the Navier–Stokes equations. To control the behavior of functions at infinity, the problem is set in weighted Sobolev spaces including anisotropic weights. In a first step, some weighted Poincare-type inequalities are obtained. In a second step, we establish existence, uniqueness and regularity results.

Backward stochastic partial differential equations in bounded and unbounded domains are studied. Existence and regularity results are obtained. Duality relationship with forward SPDEs are established. Representation of functionals of Ito processes in bounded domains is discussed.

An optimal rearrangement problem in a cylindrical domainΩ=D× (0, 1) is considered, under the constraint that the force function does not depend on thexnvariable of the cylindrical axis. This leads to a new type of obstacle problem in the cylindrical domain Δu(x′,xn) =χ{v>0}(x′) +χ{v=0}(x′) [∂νu(x′,0) +∂νu(x′, 1)]arising from minimization of the functional ∫Ω½;|∇u(x)|2+χ{v>0}(x′)u(x) dx,wherev(x′) =∫01u(x′,t)dt, and∂νuis the exterior normal derivative ofuat the boundary. Several existence and regularity results are proven and it is shown that the comparison principle does not hold for minimizers.

We prove global second derivative estimates for the Dirichlet problem for degenerate Monge-Ampere equations which yield corresponding existence and regularity results. Our conditions are essentially optimal and our techniques, while drawing on previous investigations, are substantially new.

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.

Noncoercive elliptic equations with discontinuous coefficients in unbounded domains