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article

Positive radial solutions for Dirichlet problems via a Harnack‐type inequality

Radu Precup, Jorge Rodríguez‐López

Mathematical Methods in the Applied Sciences > 46 > 2 > 2972 - 2985

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving

ϕ

‐Laplacian operators in a ball. In particular,

p

‐Laplacian and Minkowski‐curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack‐type inequality in terms of a seminorm. As a consequence of the localization result,...

article

Dependence on parameters for nonlinear equations—Abstract principles and applications

Michał Bełdziński, Marek Galewski, Igor Kossowski

Mathematical Methods in the Applied Sciences > 45 > 3 > 1668 - 1686

We provide parameter‐dependent version of the Browder–Minty Theorem in case when the solution is unique utilizing different types of monotonicity and compactness assumptions related to condition (S)₂. Potential equations and the convergence of their Euler action functionals are also investigated. Applications towards the dependence on parameters for both potential and nonpotenial nonlinear Dirichlet...

article

Two positive solutions for a Dirichlet problem with the $(p, q)$ ‐Laplacian

Angela Sciammetta, Elisabetta Tornatore

Mathematische Nachrichten > 293 > 5 > 1004 - 1013

The aim of this paper is to prove the existence of two solutions for a nonlinear elliptic problem involving the

(p, q)

‐Laplacian operator. The solutions are obtained by using variational methods and critical points theorems. The positivity of the solutions is shown by applying a generalized version of the strong maximum principle.

article

Dirichlet Problem in Parallelepiped for Elliptic Equation with Three Singular Coefficients

A. A. Abashkin, I. P. Egorova

Russian Mathematics > 2019 > 63 > 10 > 1-12

We study the Dirichlet problem in the parallelepiped for an equation that is an analogue of the generalized bi-axisymmetric Helmholtz equation for the case of three independent variables. Using spectral analysis methods, we prove the uniqueness of the solution to the problem under study. The solution is constructed in the form of a double series using expansion in a Fourier-Bessel series. Sufficient...

article

Quasianalytic Functional Classes in Jordan Domains of the Complex Plane

R. A. Gaisin

Journal of Mathematical Sciences > 2019 > 241 > 6 > 701-717

In this paper, we study the Carleman classes in Jordan domains of the complex plane. We obtain a quasianalyticity criterion for the regular Carleman classes, which is universal for all weakly uniform domains. The proof is based on the solution of the Dirichlet problem with an unbounded boundary function and a Beurling result on the estimate of the harmonic measure.

article

Inverse Problems for Initial Conditions of the Mixed Problem for the Telegraph Equation

K. B. Sabitov, A. R. Zaynullov

Journal of Mathematical Sciences > 2019 > 241 > 5 > 622-645

In this paper, we examine inverse problems for initial conditions for the wave and telegraph equations and state uniqueness criteria. Solutions of these problems are constructed in the series form. In the proof of uniform convergence of these series, the problem on small denominators appears. We prove estimates of small denominators separated from zero and obtain asymptotics that allow one to justify...

article

Brosamler’s formula revisited and extensions

Lucian Beznea, Mihai N. Pascu, Nicolae R. Pascu

Analysis and Mathematical Physics > 2019 > 9 > 2 > 747-760

Brosamler’s formula gives a probabilistic representation of the solution of the Neumann problem for the Laplacian on a smooth bounded domain $$D\subset \mathbb {R}^n$$ D ⊂ R n in terms of the reflecting Brownian motion in D. The original proof, as well as other proofs in the literature (e.g., in the case of Lipschitz domains), are based on potential theory (transition densities of the...

article

Extrapolation for the L_p Dirichlet Problem in Lipschitz Domains

Zhongwei Shen

Acta Mathematica Sinica, English Series > 2019 > 35 > 6 > 1074-1084

Let L be a second-order linear elliptic operator with complex coefficients. It is shown that if the L^p Dirichlet problem for the elliptic system L(u) = 0 in a fixed Lipschitz domain Ω in ℝ^d is solvable for some $$1 < p = {p_0} < \frac{{2\left( {d - 1} \right)}}{{d - 2}},$$ 1 < p = p 0 < 2 ( d − 1 ) d − 2 , then it is solvable for all p satisfying $${p_0} < p <...

article

The Green Function of the Dirichlet Problem for the Biharmonic Equation in a Ball

V. V. Karachik

Computational Mathematics and Mathematical Physics > 2019 > 59 > 1 > 66-81

An elementary solution of the biharmonic equation is defined. By using the properties of the Gegenbauer polynomials, series expansions of this elementary solution and an associated function with respect to a complete system of homogeneous harmonic polynomials orthogonal on a unit sphere are obtained. Then the Green function of the Dirichlet problem for the biharmonic equation in a unit ball is constructed...

article

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

Yaning Wang

Boundary Value Problems > 2019 > 2019 > 1 > 1-10

In this paper, we establish the existence of multiple positive radial solutions for the Dirichlet problem involving two mean curvature equations of spacelike graphs in Euclidean and Minkowski spaces.

article

On Fractional Analogs of Dirichlet and Neumann Problems for the Laplace Equation

Batirkhan Turmetov, Kulzina Nazarova

Mediterranean Journal of Mathematics > 2019 > 16 > 3 > 1-17

In this paper, we investigate solvability of fractional analogs of the Dirichlet and Neumann boundary-value problems for the Laplace equation. Operators of fractional differentiation in the Riemann–Liouville and Caputo sense are considered as boundary operators. The considered problems are solved by reducing them to Fredholm integral equations. Theorems on existence and uniqueness of solutions of...

article

Optimality and resonances in a class of compact finite difference schemes of high order

Joackim Bernier

Calcolo > 2019 > 56 > 2 > 1-33

In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an algebraic structure. We give some general criteria of convergence and we apply them to obtain two new results. On the one hand, we use Padé approximant theory to...

article

The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property

Tomasz Adamowicz, Nageswari Shanmugalingam

Mathematische Zeitschrift > 2019 > 293 > 3-4 > 1633-1656

Prime end boundaries $${\partial _\mathrm{P}}\Omega $$ ∂ P Ω of domains $$\Omega $$ Ω are studied in the setting of complete doubling metric measure spaces supporting a p-Poincaré inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity...

article

Existence results for Dirichlet problems about two mean curvature equations in Euclidean and Minkowski spaces

Yaning Wang

Journal of Fixed Point Theory and Applications > 2019 > 21 > 1 > 1-9

Let $$H_L$$ HL and $$H_R$$ HR be the extrinsic mean curvatures of a spacelike graph in Minkowski space $$\mathbb {L}^{N+1}$$ LN+1 and Euclidean space $$\mathbb {R}^{N+1}$$ RN+1 , respectively. In this paper, we establish the existence of multiple positive radial solutions for the Dirichlet problem of two nonlinear partial differential equations involving $$H_L$$ HL and $$H_R$$ HR .

article

Solution the Dirichlet Problem for Multiply Connected Domain Using Numerical Conformal Mapping

D. F. Abzalilov, P. N. Ivanshin, E. A. Shirokova

Complex Analysis and Operator Theory > 2019 > 13 > 3 > 1419-1429

We present a method for construction of continuous approximate 2D Dirichlet problems solutions in an arbitrary multiply connected domain with a smooth boundary. The method is based on integral equations solution which is reduced to a linear system solution and does not require iterations. Unlike the Fredholm’s solution of the problem ours applies not a logarithsmic potential of a double layer but...

article

Pseudo-differential equations and conical potentials: 2-dimensional case

Vladimir B. Vasilyev

Opuscula Mathematica > 2019 > Vol. 39, no. 1 > 109--124

We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for...

article

Optimal Recovery Methods for Solutions of the Dirichlet Problem that are Exact on Subspaces of Spherical Harmonics

E. A. Balova, K. Yu. Osipenko

Mathematical Notes > 2018 > 104 > 5-6 > 781-788

We consider the optimal recovery problem for the solution of the Dirichlet problem for the Laplace equation in the unit d-dimensional ball on a sphere of radius ρ from a finite collection of inaccurately specified Fourier coefficients of the solution on a sphere of radius r, 0 < r < ρ < 1. The methods are required to be exact on certain subspaces of spherical harmonics.

article

Radially Symmetric Solutions of the p-Laplace Equation with Gradient Terms

Ar. S. Tersenov

Journal of Applied and Industrial Mathematics > 2018 > 12 > 4 > 770-784

We consider the Dirichlet problem for the p-Laplace equation with nonlinear gradient terms. In particular, these gradient terms cannot satisfy the Bernstein—Nagumo conditions. We obtain some sufficient conditions that guarantee the existence of a global bounded radially symmetric solution without any restrictions on the growth of the gradient term. Also we present some conditions on the function simulating...

article

Ground state solutions of Kirchhoff-type fractional Dirichlet problem with p-Laplacian

Taiyong Chen, Wenbin Liu

Advances in Difference Equations > 2018 > 2018 > 1 > 1-9

We consider the Kirchhoff-type p-Laplacian Dirichlet problem containing the left and right fractional derivative operators. By using the Nehari method in critical point theory, we obtain the existence theorem of ground state solutions for such Dirichlet problem.

article

Defect Numbers of the Dirichlet Problem for a Properly Elliptic Sixth-Order Equation

A. O. Babayan, S. O. Abelyan

Mathematical Notes > 2018 > 104 > 3-4 > 339-347

The Dirichlet problem for a class of properly elliptic sixth-order equations in the unit disk is considered. Formulas for determining the defect numbers of this problem are obtained. Linearly independent solutions of the homogeneous problem and conditions for the solvability of the inhomogeneous problem are given explicitly.

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Positive radial solutions for Dirichlet problems via a Harnack‐type inequality

Dependence on parameters for nonlinear equations—Abstract principles and applications

Two positive solutions for a Dirichlet problem with the $(p, q)$ ‐Laplacian

Dirichlet Problem in Parallelepiped for Elliptic Equation with Three Singular Coefficients

Quasianalytic Functional Classes in Jordan Domains of the Complex Plane

Inverse Problems for Initial Conditions of the Mixed Problem for the Telegraph Equation

Brosamler’s formula revisited and extensions

Extrapolation for the L_p Dirichlet Problem in Lipschitz Domains

The Green Function of the Dirichlet Problem for the Biharmonic Equation in a Ball

Multiple radial solutions for Dirichlet problem involving two mean curvature equations in Euclidean and Minkowski spaces

On Fractional Analogs of Dirichlet and Neumann Problems for the Laplace Equation

Optimality and resonances in a class of compact finite difference schemes of high order

The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property

Existence results for Dirichlet problems about two mean curvature equations in Euclidean and Minkowski spaces

Solution the Dirichlet Problem for Multiply Connected Domain Using Numerical Conformal Mapping

Pseudo-differential equations and conical potentials: 2-dimensional case

Optimal Recovery Methods for Solutions of the Dirichlet Problem that are Exact on Subspaces of Spherical Harmonics

Radially Symmetric Solutions of the p-Laplace Equation with Gradient Terms

Ground state solutions of Kirchhoff-type fractional Dirichlet problem with p-Laplacian

Defect Numbers of the Dirichlet Problem for a Properly Elliptic Sixth-Order Equation

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