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In this paper, we study the Dirichlet problem of Hessian quotient equations of the form $S_k(D^2u)/S_l(D^2u)=g(x)$ in exterior domains. For $g\\equiv \\mbox {const.}$ , we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron’s method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.

We investigate the existence, uniqueness and multiplicity of one-signed rotationally symmetric solutions of singular Dirichlet problems with the prescribed higher mean curvature operator in Minkowski spacetime. The main tools are the Schauder fixed point theorem along with cut-off technique and the  Leggett-Williams fixed point theorem.  In addition, we give some practical models to illustrate the effectiveness of our results.

We propose a new monotone finite difference discretization for the variational p -Laplace operator, Δ p u = div ( | ∇ u | p - 2 ∇ u ) , and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p -Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

The purpose of the article is to investigate Dirichlet boundary-value problems of the fractional$ p $ -Laplacian equation with impulsive effects. By using the Nehari manifold method, mountain pass theorem and three critical points theorem, some new results are achieved under more general growth conditions. In addition, this paper weakens the commonly used$ p $ -suplinear and$ p $ -sublinear growth conditions.

The fourth-order properly elliptic equation with multiple root is considered in the elliptic domain. The conditions, necessary and sufficient for the unique solvability of the Dirichlet problem for this equation are found, and if these conditions fail the defect numbers of this problem are determined. The solution of the problem is found in explicit form.

We characterize generalized derivatives of the solution operator of the obstacle problem. This precise characterization requires the usage of the theory of so-called capacitary measures and the associated solution operators of relaxed Dirichlet problems. The generalized derivatives can be used to obtain a novel necessary optimality condition for the optimal control of the obstacle problem with control constraints. A comparison shows that this system is stronger than the known system of C-stationarity.

The strong solvability of the Cauchy-Dirichlet problem for parabolic quasilinear equations with discontinuous data is investigated. The leading coefficients depend on the point ( x , t ) and on the solution u , the dependence on x is of the VMO type, while with respect to t only measurability is required from them. Under suitable structural conditions on the nonlinear terms, the existence and uniqueness of a strong solution is proved, which also turns out to be Hölder continuous.

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p -Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

In an unbounded domain of RN, N≥2, we prove existence and Stampacchia type regularity of solutions to some noncoercive nonlinear Dirichlet problems whose model case appears in stationary convection–diffusion phenomena. The drift term is controlled through a function in a suitable functional space, strictly containing Lebesgue one. We obtain some a priori estimates, by contradiction, via a weak maximum principle.

In this paper, we provide a new means of establishing solvability of the Dirichlet problem on Lipschitz domains, with measurable data, for second order elliptic, nonsymmetric divergence form operators. We will show that a certain optimal Carleson measure estimate for bounded solutions of such operators implies a regularity result for the associated elliptic measure.