[Paper Review] Smoothness of generalized solutions for nonlocal elliptic problems on the plane

Pavel Gurevich|arXiv (Cornell University)|Apr 21, 2014

Differential Equations and Boundary Problems3 citations

TL;DR

This paper establishes necessary and sufficient conditions for generalized solutions of nonlocal elliptic problems on bounded planar domains to possess second-order Sobolev regularity ($W_2^2$). By analyzing singularities arising when nonlocal terms intersect the boundary, it introduces consistency conditions on nonlocal operators and right-hand sides that guarantee smoothness, using asymptotic analysis in weighted Sobolev spaces and model problems in angular domains.

ABSTRACT

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We announce conditions that are necessary and sufficient for any generalized solution to possess an appropriate smoothness (in terms of Sobolev spaces). The proofs are given in the forthcoming paper.

Motivation & Objective

• To determine conditions under which generalized solutions of nonlocal elliptic problems on bounded planar domains belong to the Sobolev space $W_2^2(G)$, despite potential singularities near boundary points.
• To address the critical challenge in nonlocal elliptic theory where nonlocal terms' supports intersect the boundary, leading to power-law singularities in solutions.
• To characterize the structure of nonlocal operators and boundary data that preserve regularity, particularly near corner points of the domain.
• To establish a framework for both homogeneous and inhomogeneous nonlocal boundary conditions, with variable coefficients near conjugation points.
• To extend results from the Dirichlet problem to general nonlocal conditions for second-order elliptic equations, with potential generalization to higher-order equations.

Proposed method

• Introduce a finite set $\mathcal{K}$ of corner points on the boundary $\partial G$, where nonlocal terms may be supported, and decompose $\partial G \setminus \mathcal{K}$ into smooth curves $\Gamma_i$.
• Define nonlocal operators $\mathbf{B}_i^1$ and $\mathbf{B}_i^2$ corresponding to nonlocal terms supported near $\mathcal{K}$, with $\mathbf{B}_i^1$ acting via diffeomorphisms $\Omega_{is}$ mapping boundary curves into the interior.
• Apply local change of variables near each $g_j \in \mathcal{K}$ to transform the domain into a plane angle $K_j = \{ r>0, |\omega| < \omega_j \}$, enabling analysis in polar coordinates.
• Introduce consistency conditions (11) on the behavior of nonlocal operators and their action on functions in weighted Sobolev spaces, ensuring regularity of solutions.
• Use model problems in angular domains and asymptotic expansions in weighted $L^2$-Sobolev spaces to derive necessary and sufficient conditions for $W_2^2$-regularity.
• Establish Condition 4 and its weaker variant Condition $4'$, which require that nonlocal operators preserve consistency of traces and derivatives at corner points, ensuring smoothness of solutions.

Experimental results

Research questions

• RQ1 Under what conditions on nonlocal operators and boundary data is a generalized solution $u \in W_2^1(G)$ of a nonlocal elliptic problem guaranteed to lie in $W_2^2(G)$?
• RQ2 How do nonlocal terms supported near boundary corner points affect the regularity of solutions, and what structural constraints prevent singularities?
• RQ3 What role do consistency conditions on the nonlocal operators—particularly those involving compositions of diffeomorphisms and their inverses—play in ensuring $W_2^2$-regularity?
• RQ4 Can the requirement for smoothness be reduced to conditions on the behavior of nonlocal operators near corner points, independent of the global domain structure?
• RQ5 How do the results for inhomogeneous nonlocal conditions compare to those for homogeneous conditions, and what is the significance of admissible functions and vectors in the homogeneous case?

Key findings

• A generalized solution $u \in W_2^1(G)$ of a nonlocal elliptic problem belongs to $W_2^2(G)$ if and only if the nonlocal operators and right-hand sides satisfy the consistency condition (11), which ensures regularity of traces and derivatives at corner points.
• Condition 4 is both necessary and sufficient for $W_2^2$-regularity of solutions with inhomogeneous nonlocal data $\{f_0, f_i\} \in L_2(G) \times \mathcal{S}_2^{3/2}(\partial G)$, where $\mathcal{S}_2^{3/2}(\partial G)$ is the space of data satisfying the consistency condition.
• If Condition 4 fails, there exists a generalized solution with data in $L_2(G) \times \mathcal{S}_2^{3/2}(\partial G)$ that does not belong to $W_2^2(G)$, proving its necessity.
• For homogeneous nonlocal conditions, Condition $4'$—a weaker version of Condition 4—serves as a necessary and sufficient condition for $W_2^2$-regularity of solutions.
• The failure of Condition $4'$ implies the existence of a generalized solution in $W_2^1(G) \setminus W_2^2(G)$, even with homogeneous nonlocal data.
• The results are derived via asymptotic analysis of model problems in angular domains and rely on the solvability of nonlocal problems in weighted Sobolev spaces, with key results from [11], [2], and [12].