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We address the numerical solution of the Dirichlet problem for the two-dimensional elliptic Monge--Ampere equation using a least-squares/relaxation approach. The relaxation algorithm allows the decoupling of the differential operators from the nonlinearities of the equation, within a splitting approach. The approximation relies on mixed low order finite element methods with regularization techniques. In order to account for data singularities in non-smooth cases, we introduce an adaptive mesh refinement technique. The error indicator is based an independent formulation of the Monge--Ampere equation under divergence form, which allows to explicit a residual term. We show that the error is bounded from above by an a posteriori error indicator plus an extra term that remains to be estimated. This indicator is then used within the existing least-squares framework. The results of numerical experiments support the convergence of our relaxation method to a convex classical solution, if such a solution exists. Otherwise they support convergence to a generalized solution in a least-squares sense. Adaptive mesh refinement proves to be efficient, robust, and accurate to tackle test cases with singularities. [ABSTRACT FROM AUTHOR]

Abstract Integral estimates for weak solutions to a class of Dirichlet problems for nonlinear, fully anisotropic, elliptic equations with a zero order term are obtained by symmetrization techniques. The anisotropy of the principal part of the operator is governed by a general n -dimensional Young function of the gradient which is not necessarily of polynomial type and need not satisfy the Δ 2 -condition. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the quasilinear elliptic problem − div ∇ u ∕ 1 + | ∇ u | 2 + a (x) u = b (x) ∕ 1 + | ∇ u | 2 in B , u = 0 on ∂ B , where B is an open ball in R N , with N ≥ 2 , and a , b ∈ C 1 (B ¯) are given radially symmetric functions, with a (x) ≥ 0 in B. This class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Unlike all previous works published on these subjects, existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients a , b are not necessarily constant and no sign condition is assumed on b. [ABSTRACT FROM AUTHOR]

We study an optimal control problem associated to a Dirichlet boundary value problem of the type (BVP) div [β(x)∇u(x) + (A x/|x|2 +g(x)) u(x)] = div F, u ∈ W1,p0(Ω),1 < p ⩽ 2, where Ω is a bounded regular domain of RN, 0 ∈ Ω, β : Ω → R is an unbounded function satisfying β(x) ⩾ λ0 > 0 a.e., A is a suitably small constant, and g ∈ L∞(Ω;RN).We consider the vector field F as the control and the corresponding weak solution u to (BVP) as the state. Our aim is to find the optimal vector field F ∈ Lp(Ω) so that the corresponding state u ∈ W1,p0(Ω) is close to the desired profile in Lp(Ω) while the norm of u in W1,p(Ω) is not too large.We prove that, for every p less than 2 and suitably close to 2, (BVP) admits an unique weak solution and for such values of p, we prove the existence of optimal pairs. [ABSTRACT FROM AUTHOR]

Several important problems in partial differential equations can be formulated as integral equations. Often the integral operator defines the solution of an elliptic problem with specified jump conditions at an interface. In principle the integral equation can be solved by replacing the integral operator with a finite difference calculation on a regular grid. A practical method of this type has been developed by the second author. In this paper we prove the validity of a simplified version of this method for the Dirichlet problem in a general domain in R2 or R3. Given a boundary value, we solve for a discrete version of the density of the double layer potential using a low order interface method. It produces the Shortley-Weller solution for the unknown harmonic function with accuracy O(h2). We prove the unique solvability for the density, with bounds in norms based on the energy or Dirichlet norm, using techniques which mimic those of exact potentials. The analysis reveals that this crude method maintains much of the mathematical structure of the classical integral equation. Examples are included. [ABSTRACT FROM AUTHOR]

A normal mode analysis of a vibrating mechanical or electrical system gives rise to an eigenvalue problem. Faber made a fairly complete study of the existence and asymptotic behavior of eigenvalues and eigenfunctions, Green's function, and expansion properties. We will investigate a new characterization of some class nonlinear eigenvalue problem. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with some homogeneous Dirichlet problems of nonlinear diffusion equations. After demonstrating the existence and uniqueness of weak solutions, we prove the existence and non-existence of global solutions. The influence of coefficients and geometry of domain is shown clearly on the existence of global solutions. We give the complete classification of simultaneous blow-up of solutions with blow-up rates as well in one-dimensional space. Moreover, the bounds of blow-up time are studied for all dimensions of space domain. [ABSTRACT FROM AUTHOR]

We propose an L2 -version of the divergence theorem. The Green and Poisson operators associated with the infinite-dimensional version of the Dirichlet problem are investigated. [ABSTRACT FROM AUTHOR]

Abstract The paper deals with a nonlinear nonlocal second-order p -Laplacian equation with Dirichlet boundary conditions. A rigorous proof for existence and uniqueness of the weak solution is presented. The weak formulation of the problem of interest is transformed into an unconstrained minimization problem. A variational inequality for the objective functional is obtained. A priori estimates for the weak solution are proved. The optimization problem is solved by means of the two-point step size gradient method. A steplength for the iterative method assuring monotone decrease of the error in approximate solutions is found. Q -linear convergence of the finite element approximations to the discrete solution is established. A monotone error reduction is proved. An original procedure for obtaining initial guesses is developed. Numerical examples supporting the developed theory are discussed. [ABSTRACT FROM AUTHOR]

It is known (see [14] ) that, for every Lipschitz domain Ω on the plane Ω = { x + i y : y > ν ( x ) } , with ν a real valued Lipschitz function, there exists 1 ≤ p 0 < 2 so that the Dirichlet problem has a solution for every function f ∈ L p ( d s ) and every p ∈ ( p 0 , ∞ ) . Moreover, if p 0 > 1 , the result is false for every p ≤ p 0 . The purpose of this paper is to study in more detail what happens at the endpoint p 0 ; that is, we want to find spaces X ⊂ L p 0 so that the Dirichlet problem is solvable for every f ∈ X . These spaces X will be either the Lorentz space L p 0 , 1 ( d s ) or some type of logarithmic Orlicz space. Our results will be applied to the special case of Schwarz–Christoffel Lipschitz domains, among others, for which we explicitly compute the value of p 0 . [ABSTRACT FROM AUTHOR]

The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this research article is to present results on the numerical simulation for time‐fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space that were found in the transonic flows. Those resulting mathematical models are solved using the reproducing kernel algorithm which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the algorithm are discussed through academic validations. [ABSTRACT FROM AUTHOR]

For a higher-order linear hyperbolic equation with nonsmooth coefficients, we consider the Dirichlet and Lidstone problems in a rectangle with nonclassical boundary conditions and prove that these problems are equivalent to the classical Dirichlet and Lidstone problems, respectively. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem for positive solutions of the equation − Δ p ( u ) = f ( u ) in a convex, bounded, smooth domain Ω ⊂ R N , with f locally Lipschitz continuous. We provide sufficient conditions guaranteeing L ∞ a priori bounds for positive solutions of some elliptic equations involving the p -Laplacian and extend the class of known nonlinearities for which the solutions are L ∞ a priori bounded. As a consequence we prove the existence of positive solutions in convex bounded domains. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem of the n-dimensional complex Monge-Ampère equation det( u ) = F/| z|, where 0 < α < n. This equation comes from La Nave-Tian's continuity approach to the Analytic Minimal Model Program. [ABSTRACT FROM AUTHOR]

In the paper, we establish commutator estimates for the Dirichlet-to-Neumann map of Stokes systems in Lipschitz domains. The approach is based on Dahlberg's bilinear estimates, and the results may be regarded as an extension of [8,21] to Stokes systems. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with modeling and analyzing the dynamics for a two-stage species that lives on a half plane. We first derive a spatially nonlocal and temporally delayed differential equation that describes the mature population on a semi-infinite environment with a homogeneous Dirichlet condition. For the derived model, we are able to show that the solutions induce a k -set contraction semiflow with respect to the compact open topology on a bounded positive invariant set attracting every solution of the equation. To describe the global dynamics, we first establish a priori estimate for nontrivial solutions after exploring the delicate asymptotic properties of the nonlocal delayed effect, which enables us to show the repellency of the trivial equilibrium. Using the estimate, k -set contracting property as well as Schauder fixed point theorem, we then establish the existence of a positive spatially heterogeneous steady state. At last, we show global attractivity of the nontrivial steady state by employing dynamical system approaches. [ABSTRACT FROM AUTHOR]

Copulas offer a very general tool to describe the dependence structure of random variables supported by the hypercube. Inspired by problems of species abundances in Biology, we study three distinct toy models where copulas play a key role. In a first one, a Marshall–Olkin copula arises in a species extinction model with catastrophe. In a second one, a quasi-copula problem arises in a flagged species abundance model. In a third model, we study completely random species abundance models in the hypercube as those, not of product type, with uniform margins and singular. These can be understood from a singular copula supported by an inflated simplex. An exchangeable singular Dirichlet copula is also introduced, together with its induced completely random species abundance vector. [ABSTRACT FROM AUTHOR]

The rate of convergence of approximate solutions via penalization for free boundary problems are concerned. A key observation is to obtain global bounds of penalized terms which give necessary estimates on integrations by the nonlinear adjoint method by L.C. Evans. [ABSTRACT FROM AUTHOR]

The behavior of solutions of the Poisson equation on noncompact Riemannian manifolds of a special form is studied. Sharp conditions for the unique solvability of the Dirichlet problem on the reconstruction of solutions of the Poisson equation from continuous boundary data at infinity are found. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three-dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in B M O norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO-type estimate for the Lam [ABSTRACT FROM AUTHOR]

We study the existence of non-trivial weak solutions in W01,p (Ω) of the super-linear Dirichlet problem -div(|∇u|p-2∇u) = f(x,u) in Ω, u = 0 on ∂Ω where f satisfies the condition |f(x, t)| ≤ |ω(x)t| r-1 + b(x) ∀(x,t) ∊ Ω × R; where r ∊ (p, Np/N-p ), b ∊ L r/r-1 (Ω) and |ω|r-1 may be non-integrable on Ω. [ABSTRACT FROM AUTHOR]

In this article we analyze some possibilities of finding positive solutions for second-order boundary-value problems with the Dirichlet and periodic boundary conditions, for which the corresponding Green's functions change sign. The obtained results can also be adapted to Neumann and mixed boundary conditions. [ABSTRACT FROM AUTHOR]

Three main results concerning the infinity-Laplacian are proved. Theorem 1.1 shows that some overdetermined problems associated to an inhomogeneous infinity-Laplace equation are solvable only if the domain is a ball centered at the origin : this is the reason why we speak of constrained radial symmetry. Theorem 1.2 deals with a Dirichlet problem for infinity-harmonic functions in a domain possessing a spherical cavity. The result shows that under suitable control on the boundary data the unknown part of the boundary is relatively close to a sphere. Finally, Theorem 1.4 gives boundary conditions implying that the unknown part of the boundary is exactly a sphere concentric to the cavity. Incidentally, a boundary-point lemma of Hopf’s type for the inhomogeneous infinity-Laplace equation is obtained. [ABSTRACT FROM AUTHOR]

We study the existence of solutions of the Dirichlet problem for the Schrödinger operator with measure data { − Δ u + V u = μ in Ω , u = 0 on ∂ Ω . We characterize the finite measures μ for which this problem has a solution for every nonnegative potential V in the Lebesgue space L p ( Ω ) with 1 ≤ p ≤ N 2 . The full answer can be expressed in terms of the W 2 , p capacity for p > 1 , and the W 1 , 2 (or Newtonian) capacity for p = 1 . We then prove the existence of a solution of the problem above when V belongs to the real Hardy space H 1 ( Ω ) and μ is diffuse with respect to the W 2 , 1 capacity. [ABSTRACT FROM AUTHOR]

A general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh–Weyl m -function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps. [ABSTRACT FROM AUTHOR]

Let a ( X , y ) be a nonnegative radical potential in a cylinder. In this paper, we study the solutions of the Dirichlet–Sch problem associated with a stationary Schrödinger operator. Existence of solutions for the Schrödinger equation and asymptotic properties as y → + ∞ and y → − ∞ are also considered under suitable conditions. [ABSTRACT FROM AUTHOR]

We study a system of partial differential equations describing a steady thermoconvective ow of a non-Newtonian uid. We assume that the stress tensor and the heat ux depend on temperature and satisfy the conditions of p, q-coercivity with p > 2n/n+2, q > np/p(n+1)-n, respectively. Considering Dirichlet boundary conditions for the velocity and a mixed and nonlinear boundary condition for the temperature, we prove the existence of weak solutions. We also analyze the existence and uniqueness of strong solutions for small and suitably regular data. [ABSTRACT FROM AUTHOR]

This article concerns the existence of sign-changing solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions, LKu = f(x, u), x ∊ Ω, u = 0; x ∊ Rn\Ω where Ω ⊂ Rn (n≥2) is a bounded, smooth domain and the nonlinear term f satisfies suitable growth assumptions. By using Brouwer's degree theory and Deformation Lemma and arguing as in [2], we prove that there exists a least energy sign-changing solution. Our results generalize and improve some results obtained in [27]. [ABSTRACT FROM AUTHOR]

We study the existence and multiplicity of solutions of the problem -Δu = λ1u- + g (x, u) in Ω; u = 0, on φΩ. where is a smooth bounded domain in RN (N ≥ 2), u- denotes the negative part of u: Ω R, ≥1 is the first eigenvalue of the N-dimensional Laplacian with Dirichlet boundary conditions in, and g : Ω R × R is a continuous function with g(x, 0) = 0 for all ∊ Ω. We assume that the nonlinearity g(x, s) has a strong resonant behavior for large negative values of s and is superlinear, but subcritical, for large positive values of s. Because of the lack of compactness in this kind of problem, we establish conditions under which the associated energy functional satisfies the Palais-Smale condition. We prove the existence of three nontrivial solutions of problem (1) as a consequence of Ekeland's Variational Principle and a variant of the mountain pass theorem due to Pucci and Serrin [14]. [ABSTRACT FROM AUTHOR]

In this article, we use Morse theory to investigate a type of Dirichlet boundary value problem related to the (2, p)-Laplacian operator, where the nonlinear term is characterized by the first eigenvalue of the Laplace operator. The investigation is heavily based on a new decomposition about the Banach space W¹0,p(Ω), where ⊂ RN (N ≥ 1) is a bounded domain with smooth enough boundary. [ABSTRACT FROM AUTHOR]

We consider a Dirichlet problem driven by the sum of a p -Laplacian and a Laplacian (known as a ( p , 2 ) -equation) and with a nonlinearity which exhibits asymmetric behavior as s → ± ∞ . More precisely, it is ( p − 1 ) -superlinear near +∞ (but without satisfying the Ambrosetti–Rabinowitz condition) and it is ( p − 1 ) -sublinear near −∞ and possibly resonant with respect to the principal eigenvalue of the p -Laplacian. Using variational tools along with Morse theory we prove a multiplicity theorem generating five nontrivial solutions (one is negative, two are positive, one is nodal and for the fifth we do not have any information about its sign). [ABSTRACT FROM AUTHOR]

In this paper we consider the following nonlinear subelliptic Dirichlet problem: where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth vector fields defined in $\mathbf{R}^{n}$ with globally Lipschitz coefficients satisfying Hörmander's condition, and we prove the global higher integrability for the very weak solutions. [ABSTRACT FROM AUTHOR]

We give an explicit upper bound for the number of zeros of Hecke–Landau zeta-functions in a rectangle. [ABSTRACT FROM PUBLISHER]

Let U be a bounded open connected set in $$\mathbb {R}^n$$ ( $$n\ge 1$$ ). We refer to the unique weak solution of the Poisson problem $$-\Delta u = \chi _A$$ on U with Dirichlet boundary conditions as $$u_A$$ for any measurable set A in U. The function $$\psi :=u_U$$ is the torsion function of U. Let V be the measure $$V:=\psi \,\mathscr {L}^n$$ on U where $$\mathscr {L}^n$$ stands for n-dimensional Lebesgue measure. We study the variational problem with $$p\in (0,1)$$ where $$J(A):=\int _Au_A\,dx$$ and the supremum is taken over measurable sets $$A\subset U$$ subject to the constraint $$V(A)=pV(U)$$ . We relate the above problem to an unstable two-phase membrane problem. We characterise optimsers in the case $$n=1$$ . The proof makes use of weighted isoperimetric and Pólya-Szegö inequalities. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for the solvability of the Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain. The solution is represented by a convergent series constructed from the problem data. Some cases of nonuniqueness of the solution are described. [ABSTRACT FROM AUTHOR]

We design here a finite-dimensional feedback stabilising Dirichlet boundary controller for the equilibrium solutions to parabolic equations. These results extend that ones in Barbu (2013), which provide a feedback controller expressed in terms of the eigenfunctions φjcorresponding to the unstable eigenvaluesof the operator corresponding to the linearised equation. In Barbu (2013), the stabilisability result is conditioned by the require of linear independence of, on the part of the boundary where control acts. In this work, we design a similar control as in Barbu (2013), and show that it assures the stability of the system. This time, we drop the require of linear independence and any other additional hypothesis. Some examples are provided in order to illustrate the acquired results. More exactly, boundary stabilisation of the heat equation and the Fitzhugh–Nagumo equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the Dirichlet problem for degenerate parabolic equations of the form ∂ u ∂ t − div ( a ( x , t , u , ∇ u ) ) − div ϕ ( u ) = f − div g , where the main operator a ( x , t , u , ∇ u ) is allowed to degenerate with respect to the unknown u . Under suitable conditions on a and ϕ , we prove that there exists an unique solution u of this problem. [ABSTRACT FROM AUTHOR]

This work is devoted to a numerical algorithm based on the method of fundamental solutions (MFS) for solving two backward parabolic problems with different boundary conditions, one with nonlocal Dirichlet boundary conditions, and second one with Robin type boundary conditions. The initial temperature distribution will be identified from the final temperature distribution, which appear in some applied subjects. The Tikhonov regularization method with the L-curve criterion for choosing the regularization parameter is adopted for solving the resulting matrix equation which is highly ill-conditioned. Two numerical examples are provided to show the high efficiency of the suggested method. [ABSTRACT FROM PUBLISHER]

We establish the variable exponent Morrey spaces L p ( ⋅ ) , λ ( ⋅ ) estimate to the Dirichlet problem for fully nonlinear elliptic equations on a C 1 , 1 bounded domain for variable exponents p ( ⋅ ) and λ ( ⋅ ) . [ABSTRACT FROM AUTHOR]

We consider a strongly resonant homogeneous Dirichlet problem for elliptic-type equations with discontinuous nonlinearity in the phase variable. Using the variational method, we prove an existence theorem for at least three nontrivial solutions of the problem under consideration; at least two of these are semiregular. The resulting theorem is applied to the eigenvalue problem for elliptic-type equations with discontinuous nonlinearity with positive spectral parameter. An example of a discontinuous nonlinearity satisfying all the assumptions of the theorem is given. [ABSTRACT FROM AUTHOR]

We obtain improved accuracy estimates of the finite-difference scheme for a two-dimensional parabolic equation in a unit square, considering the effect of the Dirichlet boundary condition and the initial condition. We prove that the accuracy order is higher near the sides and the bottom of the space-time parallelepiped. [ABSTRACT FROM AUTHOR]

In this article, we discuss a numerical method for solving some nonlinear inverse parabolic partial differential equations with Dirichlet's boundary conditions. The approach used, is based on collocation of cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B-splines for spatial variable and derivatives, which produce an ill-posed system. We solve this system using the Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on two test problems. The figures and comparisons have been presented for clarity. Also the stability of this method has been discussed. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 88-104, 2017 [ABSTRACT FROM AUTHOR]

For a strongly elliptic system of second-order equations of a special form, formulas for the Poisson integral and Green's function in a circle and an ellipse are obtained. The operator under consideration is represented by the sum of the Laplacian and a residual part with a small parameter, and the solution to the Dirichlet problem is found in the form of a series in powers of this parameter. The Poisson formula is obtained by the summation of this series. [ABSTRACT FROM AUTHOR]

A model of measuring the level of a viscous incompressible liquid in a tank as based on the liquid level in a measuring tube is investigated. The tank is in the field of gravity, and the tank liquid level varies according to some law. As a result, a Dirichlet boundary value problem for a nonlinear integrodifferential equation of parabolic type is obtained. A global existence and uniqueness theorem is proved for a weak solution of the problem. In the case of a tank level decreasing linearly with time, it is shown numerically that the liquid level in the measuring tube oscillates with a decaying amplitude with respect to the tank level. [ABSTRACT FROM AUTHOR]