Higher Order Elliptic Equations Research Articles

Higher order elliptic equations in weighted Banach spaces

AbstractWe consider m-th order linear, uniformly elliptic equations $$\mathcal {L}u=f$$ L u = f with non-smooth coefficients in Banach–Sobolev spaces $$W_{X_w}^m (\Omega )$$ W X w m ( Ω ) generated by weighted Banach Function Spaces (BFS) $$X_w (\Omega )$$ X w ( Ω ) on a bounded domain $$\Omega \subset {\mathbb R}^{n}$$ Ω ⊂ R n . Supposing boundedness of the Hardy–Littlewood Maximal operator and the Calderón–Zygmund singular integrals in $$X_w (\Omega )$$ X w ( Ω ) we obtain solvability in the small in $$W_{X_w}^m (\Omega )$$ W X w m ( Ω ) and establish interior Schauder type a priori estimates. These results will be used in order to obtain Fredholmness of the operator $$\mathcal {L}$$ L in $$X_w (\Omega )$$ X w ( Ω ) .

Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain

Abstract This article is devoted to a global Calderón-Zygmund estimate in the framework of Lorentz spaces for the m m -order gradients of weak solution to a higher-order elliptic equation with p p -growth. We prove the main result based on a proper power decay estimation of the upper-level set by the principle of layer cake representation for the L γ , q {L}^{\gamma ,q} -estimate of D m u {D}^{m}u , while the coefficient satisfies a small BMO semi-norm and the boundary of underlying domain is flat in the sense of Reifenberg. In particular, a tricky ingredient is to establish the normal component of higher derivatives controlled by the horizontal component of higher derivatives of solutions in the neighborhood at any boundary point, which is achieved by comparing the solution under consideration with that for some reference problems.

On the Solvability of the Generalized Neumann Problem for a Higher-Order Elliptic Equation in an Infinite Domain

On the Solvability of the Generalized Neumann Problem for a Higher-Order Elliptic Equation in an Infinite Domain

Compactness of Solutions to Higher-Order Elliptic Equations

Abstract We use blow up analysis for local integral equations to prove compactness of solutions to higher-order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoen’s Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow up points of solutions.

On Local Solvability of Higher Order Elliptic Equations in Rearrangement Invariant Spaces

On Local Solvability of Higher Order Elliptic Equations in Rearrangement Invariant Spaces

On solvability in the small and Schauder-type estimates for higher order elliptic equations in grand Sobolev spaces (nonseparable case)

In this work, it is considered an elliptic operator L of mth order with nonsmooth coefficients in a non-standard grand Sobolev space on a bounded domain generated by the norm of the grand Lebesgue space . Under weaker restrictions on the coefficients of the operator, we prove the solvability (in the strong sense) in the small in and also establish interior Schauder-type estimates for these spaces. These estimates play the main role in establishing the Fredholmness of the Dirichlet problem for the equation Lu = f. The considered spaces are not separable, infinitely differentiable functions are not dense in them, and therefore many classical methods concerning Sobolev spaces are not applicable in this case. Nevertheless, it is possible to obtain the corresponding results under the assumption that the coefficients of the principal terms of the operator L are continuous, and the rest are essentially bounded in Ω.

A review on arbitrarily regular conforming virtual element methods for second- and higher-order elliptic partial differential equations

The virtual element method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order [Formula: see text], for any integer [Formula: see text]. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of [Formula: see text], [Formula: see text] being the computational domain and [Formula: see text] another suitable integer number. In this review, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation spaces with a global high-order regularity on [Formula: see text]. Then, we illustrate specific examples in the case of second- and fourth-order partial differential equations, that correspond to the cases [Formula: see text] and [Formula: see text], respectively. Finally, we investigate numerically the effect on the approximation properties of the conforming highly-regular method that results from different choices of the degree of continuity of the underlying virtual element spaces and how different stabilization strategies may impact on convergence.

On the Solvability of the Generalized Neumann Problem for a Higher-Order Elliptic Equation in an Infinite Domain

We consider the generalized Neumann problem for a 2lth-order elliptic equation with constant real higher-order coefficients in an infinite domain containing the exterior of some circle and bounded by a sufficiently smooth contour. It consists in specifying of the (kj-1){(k_j - 1)}th-order normal derivatives where 1≤k1...kl≤2l{1\le k_1 < ... < k_l \le 2l}; for kj=j{k_j = j} it turns into the Dirichlet problem, and for kj=j+1{k_j = j +1} into the Neumann problem. Under certain assumptions about the coefficients of the equation at infinity, a necessary and sufficient condition for the Fredholm property of this problem is obtained and a formula for its index in Holder spaces is given.

Monotonicity Formula and Classification of Stable Solutions to Polyharmonic Lane–Emden Equations

Abstract In this paper, we consider polyharmonic Lane–Emden equations $$ \begin{equation*} (-\Delta )^m u=|u|^{p-1}u, \ \ \ \mbox{in} \ \ \ {\mathbb R}^n, \end{equation*}$$where $m\geq 3$. We classify the stable or stable outside a compact set solutions when $m=3$ or $4$ for any dimensions and when $m\geq 5$ for large dimensions. In the process, we exhibit the general Joseph–Lundgren exponent (including both local and nonlocal cases) in a concise form and prove related properties. The key ingredient of the proof of the classification is a monotonicity formula for general polyharmonic equations, which may have application in regularity theory for higher-order elliptic equations.

The Neumann problem for higher order elliptic equations

We solve the Neumann problem in the half space for higher order elliptic differential equations with variable self-adjoint t-independent coefficients, and with boundary data in the negative smoothness space where Our arguments are inspired by an argument of Shen and build on known well posedness results in the case p = 2. We use the same techniques to establish nontangential and square function estimates on layer potentials with inputs in Lp or for a similar range of p, based on known bounds for p near 2; in this case we may relax the requirement of self-adjointess.

Remarks on radial symmetry and monotonicity for solutions of semilinear higher order elliptic equations

<abstract><p>Half a century after the appearance of the celebrated paper by Serrin about overdetermined boundary value problems in potential theory and related symmetry properties, we reconsider semilinear polyharmonic equations under Dirichlet boundary conditions in the unit ball of $ \mathbb{R}^{n} $. We discuss radial properties (symmetry and monotonicity) of positive solutions of such equations and we show that, in <italic>conformal dimensions</italic>, the associated Green function satisfies elegant reflection and symmetry properties related to a suitable Kelvin transform (inversion about a sphere). This yields an alternative formula for computing the partial derivatives of solutions of polyharmonic problems. Moreover, it gives some hints on how to modify a counterexample by Sweers where radial monotonicity fails: we numerically recover strict radial monotonicity for the biharmonic equation in the unit ball of $ \mathbb{R}^{4} $.</p></abstract>

A positivity preserving property result for the biharmonic operator under partially hinged boundary conditions

It is well known that for higher order elliptic equations, the positivity preserving property (PPP) may fail. In striking contrast to what happens under Dirichlet boundary conditions, we prove that the PPP holds for the biharmonic operator on rectangular domains under partially hinged boundary conditions, i.e., nonnegative loads yield positive solutions. The result follows by fine estimates of the Fourier expansion of the corresponding Green function.

The spike layer solution to singular perturbation Robin boundary value problem for higher order elliptic equation

The singularly perturbed Robin boundary value problem for the higher order elliptic equation is considered. Under suitable conditions, the existence and asymptotic behavior of solution to the boundary value problems are studied. The uniform validity of its asymptotic expansion is proved by using the fixed point theorem.

On solvability in the small of higher order elliptic equations in grand-Sobolev spaces

ABSTRACT This work deals with the mth order elliptic equation with non-smooth coefficients in grand-Sobolev space generated by the norm of the grand-Lebesgue space . These spaces are non-separable, and therefore, to use classical methods for treating solvability problems in these spaces, you need to modify these methods. To this aim, we consider some subspace, where the infinitely differentiable functions are dense. Then we prove that this subspace is invariant with respect to the singular integral operator and with respect to the multiplication operator by a function from . Finally, using classical method of parametrics, we prove the existence in the small of the solution to the considered equation in .

Singular Integral Operators and Elliptic Boundary-Value Problems. Part I

The monograph consists of three parts. Part I is presented here. In this monograph, we develop a new approach (mainly based on papers of the author). Many results are published for the first time here. Chapter 1 is introductory. It provides the necessary background from functional analysis (for completeness). In this monograph, we mostly use weighted HOlder spaces; they are considered in Chap. 2. Chapter 3 plays the key role: in weighted HOlder spaces, we consider estimates of integral operators with homogeneous difference kernels, covering potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Chap. 4, similar estimates in weighted Lebesgue spaces are proved. Integrals with homogeneous difference kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of first-order elliptic systems in terms of fundamental matrices or their parametrices. The investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to first-order elliptic systems.

On the generalized [formula omitted] classes of De Giorgi-Ladyzhenskaya-Ural'tseva and pointwise estimates of solutions to high-order elliptic equations via Wolff potentials

We consider quasilinear elliptic 2m-order (m⩾2) partial differential equations which prototype is∑|α|=m(−1)|α|Dα(|Dmu|p−2Dαu)=f(x),x∈Ω, where Ω is a bounded open set in Rn, n=mp and f∈L1(Ω). Using an analogue of the Kilpeläinen-Malý method, we obtain the local boundedness and continuity of arbitrary weak solution u∈Wm,p(Ω) via the Wolff potential Wm,pf.

Higher-Order Elliptic and Parabolic Equations with VMO Assumptions and General Boundary Conditions

Abstract We prove mixed L p ( L q ) -estimates, with p , q ∈ ( 1 , ∞ ) , for higher-order elliptic and parabolic equations on the half space R + d + 1 with general boundary conditions which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients which are in the class of vanishing mean oscillations both in the time variable and the space variable. In the proof, we apply and extend the techniques developed by Krylov [24] as well as Dong and Kim in [13] to produce mean oscillation estimates for equations on the half space with general boundary conditions.

Higher-Order Parabolic Equations with VMO Assumptions and General Boundary Conditions with Variable Leading Coefficients

AbstractWe prove weighted mixed $L_{p}(L_{q})$-estimates, with $p,q\in (1,\infty )$, and the corresponding solvability results for higher-order elliptic and parabolic equations on the half space ${\mathbb{R}}^{d+1}_{+}$ and on general $C^{2m-1,1}$ domains with general boundary conditions, which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients that are in the class of vanishing mean oscillations both in the time and the space variables and that the boundary operators have variable leading coefficients. The proofs are based on and generalize the estimates recently obtained by the authors in [6].

Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients

Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients

Higher-order elliptic and parabolic equations with VMO assumptions and general boundary conditions

We prove mixed Lp(Lq)-estimates, with p,q∈(1,∞), for higher-order elliptic and parabolic equations on the half space R+d+1 with general boundary conditions which satisfy the Lopatinskii–Shapiro condition. We assume that the elliptic operators A have leading coefficients which are in the class of vanishing mean oscillations both in the time variable and the space variable. In the proof, we apply and extend the techniques developed by Krylov [24] as well as Dong and Kim in [13] to produce mean oscillation estimates for equations on the half space with general boundary conditions.