[Paper Review] Exact constants in Poincare type inequalities for functions with zero mean boundary traces

Alexander I. Nazarov, Sergey Repin|arXiv (Cornell University)|Nov 9, 2012

Mathematical Approximation and Integration2 references4 citations

TL;DR

This paper derives exact, computable constants in Poincaré-type inequalities for functions with zero mean on part or all of the boundary of Lipschitz domains, focusing on rectangles, cubes, and right triangles. Using variational methods and eigenvalue analysis, the authors establish that the sharp constants are the inverse square roots of the smallest positive eigenvalues of specific Steklov-type boundary value problems, enabling precise error estimates in PDE analysis and numerical methods.

ABSTRACT

In the paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We derive exact and easily computable constants for some basic domains (rectangles, cubes, and right triangles). In the last section, we derive an a estimate of the difference between the exact solutions of two boundary value problems. Constants in Poincare type inequalities enter these estimates, which provide guaranteed a posteriori error control.

Motivation & Objective

• To determine exact, computable constants in Poincaré-type inequalities for functions with zero mean on a subset or entire boundary of a domain.
• To identify the sharp constants in inequalities involving the $ L^2 $-norm of functions and the $ L^2 $-norm of their gradients under zero mean boundary conditions.
• To apply these exact constants to derive quantitative error estimates in the analysis of elliptic partial differential equations.
• To provide explicit formulas for the best constants in basic geometric domains like rectangles, cubes, and right triangles.

Proposed method

• The authors use variational principles to show that the extremal functions in the inequalities are eigenfunctions of a specific Steklov-type eigenvalue problem.
• The sharp constant in the inequality $ \|w\|_{2,\Omega} \leq C_1(\Omega,\Gamma)\|\nabla w\|_{2,\Omega} $ is shown to be $ \lambda^{-1/2} $, where $ \lambda $ is the smallest positive eigenvalue of the problem $ -\Delta u = \lambda u $ in $ \Omega $, with Neumann conditions on $ \partial\Omega \setminus \Gamma $ and a Robin-type condition on $ \Gamma $.
• For the trace inequality $ \|w\|_{2,\Gamma} \leq C_2(\Omega,\Gamma)\|\nabla w\|_{2,\Omega} $, the sharp constant is also $ \lambda^{-1/2} $, where $ \lambda $ is the smallest positive eigenvalue of the problem $ \Delta u = 0 $ in $ \Omega $, with $ \partial_n u = \lambda u $ on $ \Gamma $ and $ \partial_n u = 0 $ on $ \partial\Omega \setminus \Gamma $.
• Explicit eigenfunctions are constructed via separation of variables for rectangular domains and cubes when $ \Gamma $ is a face.
• For right triangles, the authors use analytical and geometric arguments to identify the minimizers of the Rayleigh quotient, proving that the constructed eigenfunctions correspond to the smallest eigenvalue.
• The results are applied to derive a priori error estimates for elliptic PDEs with simplified source terms and boundary conditions, using the exact constants in a residual-based error bound.

Experimental results

Research questions

• RQ1 What are the exact values of the best constants in Poincaré-type inequalities for functions with zero mean on a measurable part of the boundary of a Lipschitz domain?
• RQ2 How can the sharp constants in these inequalities be computed explicitly for basic geometric domains such as rectangles, cubes, and right triangles?
• RQ3 What is the relationship between the sharp constants and the smallest positive eigenvalues of specific Steklov-type boundary value problems?
• RQ4 How do these exact constants improve the accuracy of a priori error estimates in the numerical solution of elliptic PDEs?
• RQ5 Can the eigenfunctions of the relevant eigenvalue problems be explicitly constructed or characterized in non-rectangular domains like right triangles?

Key findings

• For rectangles and cubes with $ \Gamma $ being one face, the exact constant in the inequality $ \|w\|_{2,\Omega} \leq C_1(\Omega,\Gamma)\|\nabla w\|_{2,\Omega} $ is $ \lambda^{-1/2} $, where $ \lambda $ is the smallest eigenvalue of the mixed Neumann-Robin problem, and this value is explicitly computable via separation of variables.
• In the case of right triangles, the authors derive the exact value of the sharp constant $ C_1(\Omega,\Gamma) $ by proving that a constructed eigenfunction corresponds to the minimal eigenvalue using geometric and analytical arguments.
• For the trace inequality $ \|w\|_{2,\Gamma} \leq C_2(\Omega,\Gamma)\|\nabla w\|_{2,\Omega} $, the sharp constant is also $ \lambda^{-1/2} $, with $ \lambda $ being the smallest positive eigenvalue of the Steklov-type problem with $ \Delta u = 0 $ and $ \partial_n u = \lambda u $ on $ \Gamma $.
• The paper provides a general error estimate for elliptic PDEs with simplified data: $ |\!|\!|u - \widehat{u}|\!|\!| \leq \rho_1 + \sqrt{\rho_2 + \rho_1^2} $, where $ \rho_1 $ and $ \rho_2 $ depend on the exact Poincaré constants and data oscillations.
• When the exact solution and its simplified counterpart share the same Dirichlet data, the error bound simplifies to $ |\!|\!|u - \widehat{u}|\!|\!| \leq \frac{D_1 + D_2}{\sqrt{c}} $, with $ D_1, D_2 $ related to data oscillations and $ c $ a coercivity constant.
• The authors demonstrate that the exact constants significantly improve the sharpness of error estimates in numerical analysis, especially when replacing complex source terms and boundary conditions with piecewise constant or affine approximations.