Elliptic Equation Of Order 2m Research Articles

Global bifurcation of positive radial solutions of elliptic equations of order 2m in annular domains

In this paper, we study the global bifurcation of positive radial solutions for some semilinear elliptic problems of order 2m in an annulus with Dirichlet boundary conditions.

Local solvability of elliptic equations of even order with Hölder coefficients

We consider elliptic equations of order 2m with H¨older coefficients. We show local solvability of the Dirichlet problem with m conditions on the boundary of the upper half space. First we consider local solvability in free space and then we treat the boundary case. Our method is based on applying the operator to an approximate solution and iterating in the Holder spaces. A priori estimates for the approximate solution is the essential part of the paper.

Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<\lambda_k\to 0$. Assuming that $$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ we prove that $\Lambda$ is an integer multiple of $\Lambda_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard $2m$-dimensional sphere.

Generalized boundary values of the solutions of semilinear elliptic equations from weighted functional spaces

In weighted C-spaces, we establish the solvability of a boundary-value problem for a semilinear elliptic equation of order 2m in a bounded domain with generalized functions given on its boundary, strong power singularities at some points of the boundary, and finite orders of singularities on the entire boundary. The behavior of the solution near the boundary of the domain is analyzed.

Generalized boundary values of solutions of quasilinear elliptic equations with linear principal part

We establish conditions for the nonlinear part of a quasilinear elliptic equation of order 2m with linear principal part under which a solution regular inside a domain and belonging to a certain weighted L1-space takes boundary values in the space of generalized functions.

On the stability of the index of unbounded nonlocal operators in Sobolev spaces

Unbounded operators corresponding to nonlocal elliptic problems on a bounded region G ⊂ ℝ2 are considered. The domain of these operators consists of functions in the Sobolev space W2m(G) that are generalized solutions of the corresponding elliptic equation of order 2m with the right-hand side in L2(G) and satisfy homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved that lower order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal perturbations on the boundary do not change the index are also formulated.

Theory of differential equations in discontinuous piecewise‐defined functions

AbstractA truly general and systematic theory of finite element methods (FEM) should be formulated using, as trial and test functions, piecewise‐defined functions that can be fully discontinuous across the internal boundary, which separates the elements from each other. Some of the most relevant work addressing such formulations is contained in the literature on discontinuous Galerkin (dG) methods and on Trefftz methods. However, the formulations of partial differential equations in discontinuous functions used in both of those fields are indirect approaches, which are based on the use of Lagrange multipliers and mixed methods, in the case of dG methods, and the frame, in the case of Trefftz method. This article addresses this problem from a different point of view and proposes a theory, formulated in discontinuous piecewise‐defined functions, which is direct and systematic, and furthermore it avoids the use of Lagrange multipliers or a frame, while mixed methods are incorporated as particular cases of more general results implied by the theory. When boundary value problems are formulated in discontinuous functions, well‐posed problems are boundary value problems with prescribed jumps (BVPJ), in which the boundary conditions are complemented by suitable jump conditions to be satisfied across the internal boundary of the domain‐partition. One result that is presented in this article shows that for elliptic equations of order 2m, with m ≥ 1, the problem of establishing conditions for existence of solution for the BVPJ reduces to that of the “standard boundary value problem,” without jumps, which has been extensively studied. Actually, this result is an illustration of a more general one that shows that the same happens for any differential equation, or system of such equations that is linear, independently of its type and with possibly discontinuous coefficients. This generality is achieved by means of an algebraic framework previously developed by the author and his collaborators. A fundamental ingredient of this algebraic formulation is a kind of Green's formulas that simplify many problems (some times referred to as Green‐Herrera formulas). An important practical implication of our approach is worth mentioning: “avoiding the introduction of the Lagrange multipliers, or the ‘frame’ in the case of Trefftz‐methods, significantly reduces the number of degrees of freedom to be dealt with.” © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Generalized Solutions of Nonlocal Elliptic Problems

An elliptic equation of order 2m with general nonlocal boundary-value conditions, in a plane bounded domain G with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space W 2 m (G) are studied. The Fredholm property of the unbounded operator (corresponding to the elliptic equation) acting on L 2(G), and defined for functions from the space W 2 m (G) that satisfy homogeneous nonlocal conditions, is established.

Estimates for the Green function and singular solutions for polyharmonic nonlinear equation

We establish a new form of the3Gtheorem for polyharmonic Green function on the unit ball ofℝn(n≥2)corresponding to zero Dirichlet boundary conditions. This enables us to introduce a new class of functionsKm,ncontaining properly the classical Kato classKn. We exploit properties of functions belonging toKm,nto prove an infinite existence result of singular positive solutions for nonlinear elliptic equation of order2m.

Solvability of nonlocal elliptic problems in weight spaces

In this paper we consider elliptic equations of order 2m in a bounded domainQ є Rn with boundaryδQ and nonlocal conditions relating the traces of the solution and its derivatives on (n − 1)-dimensional smooth manifolds Γi (∪i =∂δQ) to their values on some compact setF ⊂Q, whereF ∩δQ ≠ Φ. The Fredholm solvability of these problems in the weight spacesVp, a/l+2m(Q) is proved for arbitrary 1<p <∞.

Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C∞(S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.

A priori estimates for some semilinear elliptic equations of order 2m

A priori estimates for some semilinear elliptic equations of order 2m

Elliptic Equations of Order 2m in Annular Domains

Elliptic Equations of Order 2m in Annular Domains

Elliptic equations of order 2𝑚 in annular domains

In this paper we study the existence of positive radial solutions for some semilinear elliptic problems of order 2 m 2m in an annulus with Dirichlet boundary conditions. We consider a nonlinearity which is either sublinear or the sum of a sublinear and a superlinear term.

Positive radial solutions of semilinear elliptic equations of order 2m in annular domains

We study the existence of positive radial solutions of (-1)^{m} \Delta^{m}u=g(|x|)f(u) in an annulus with Dirichlet boundary conditions. In par- ticular L^{\infty} a priori bounds are obtained.

Comparison Theorems for Linear Elliptic Equations

AbstractTWO comparison theorems, one of pointwise type and one of integral type, will be obtained for linear elliptic equations of order 2m on an exterior domain in Rn

On the stability of index of nonlocal elliptic problems

One of the first publications devoted to nonlocal elliptic problems was the paper of T. Carleman [S]. Together with this paper are closely associated further investigations of singular integral equations with shifts and elliptic boundary value problems with shifts which map the boundary onto itself [9]. Bibliographies of publications and the theory of abstract nonlocal elliptic problems are contained in [4, 71. In [2] the Laplace equation was considered with boundary conditions connecting the values of the unknown function on a manifold r, c aQ (Q c R”) with its values on some manifold o(T, ) c Q and Dirichlet conditions on 13Q\f,. The solvability of this problem in various functional spaces was studied in [16-18,201 and elsewhere. In [ 17,201 it was shown that in the case o(F,) n (r,\r,) # 0 the Dirichlet problem with arbitrary small coefficients in nonlocal terms can have a negative index. In [lo] the formula of index for some nonlocal elliptic boundary value problems of second order in a plane domain was obtained. In the present paper we consider elliptic equations of order 2m in a bounded domain Q c R with nonlocal conditions connecting the traces of the unknown function and its derivatives on the manifolds rj (IJ, F, = 8Q) with its traces on some compacturn a c Q. In the case a c Q (Section 2) and in the case Sz n K = 0 (Section 3) we prove the stability of index of nonlocal problems in Sobolev spaces and weighted spaces, where Xi = Uj (T,\r,). In particular the set D can consist of one or several manifolds (see [2, 163). In Section 4 it is established that in the case Q n Xi = 0 index of the equation of second order, with Dirichlet conditions containing nonlocal perturbations, is equal to zero. The elliptic problems with nonlocal conditions are closely associated with boundary value problems for elliptic functional differential equations [ 171. The theory of functional differential equations for the functions of one variable was reflected in the publications most completely [IS, 131. The 323 0022-247X/91 $3.00

ELLIPTIC PROBLEMS WITH NONLOCAL CONDITIONS NEAR THE BOUNDARY

An elliptic equation of order 2m with nonlocal conditions near the boundary is considered. The presence of nonlocal terms in the boundary conditions leads to the appearance of singularities in the solution. The Noether property is proved for this problem in weighted spaces and asymptotics of the solutions are obtained. These results are applied to the study of the solvability and smoothness of solutions of elliptic differential-difference equations.Bibliography: 22 titles.

Hybrid methods from a new perspective

Usually, by a hybrid method it is understood one in which continuity requirements are reduced or eliminated altogether [Babuska, et al., 1977, 1978; Pian & Tong, 1969; Tong, 1970] by introducing auxiliary dependent variables. Finite element methods for elliptic equations of order 2m, are said to be nonconforming when in the evaluation of the energy, which involves derivatives of order m, the approximations to derivatives of order m-1 may have simple discontinuities [Nitsche, 1974; Strang, 1972].

Nonlinear elliptic equations of order 2m and subdifferentials

LetA be an operator of the calculus of variations of order 2m onWm,p(Ω) andj a normal convex integrand. Forf ∈Lp(Ω), the equation $$\mathcal{A}u + \partial j(x,u) \ni f, in \Omega , u - \phi \in W_0^{m,p} (\Omega ),$$ may have no strong solutions whenm>1, even ifj is independent ofx and φ=0. However, we obtain existence results whenj is everywhere finite and $$\int_\Omega {j(x,\phi ) dx< + \infty ,} $$ by the study of the subdifferential of the function $$\upsilon \mapsto \int_\Omega {j(x,\upsilon + \phi ) dx on W_0^{m,p} (\Omega ).} $$