[Paper Review] Isoperimetric Inequalities and Sharp Estimate for Positive Solution of Sublinear Elliptic Equations

Qiuyi Dai, Renchu He+1 more|arXiv (Cornell University)|Mar 19, 2010

Nonlinear Partial Differential Equations16 references3 citations

TL;DR

This paper establishes sharp isoperimetric estimates for positive solutions of sublinear elliptic equations using Schwarz symmetrization and rearrangement techniques. It proves that the $ L^{q+1} $-norm and essential supremum of the solution are bounded above by expressions involving the solution on a ball of equal volume, with equality if and only if the domain is a ball, providing a sharp estimate via symmetrization and comparison with a model problem on a ball.

ABSTRACT

In this paper, we prove some isoperimetric inequalities and give a sharp bound for the positive solution of sublinear elliptic equations.

Motivation & Objective

• To derive sharp upper bounds for the $ L^{k} $-norm and essential supremum of positive solutions to sublinear elliptic equations in bounded domains.
• To establish isoperimetric-type inequalities for solutions of $ -\Delta u = u^q $ with $ 0 < q < 1 $, extending eigenfunction-type symmetrization methods to nonlinear problems.
• To prove that the optimal bounds are achieved if and only if the domain $ \Omega $ is a ball, thus characterizing extremality via symmetry.
• To generalize techniques from eigenfunction symmetrization (e.g., Chiti's work) to the context of positive solutions of sublinear equations.

Proposed method

• Use of Schwarz symmetrization to transform the original domain $ \Omega $ into a ball $ \Omega^* $ of equal volume, preserving measure and enabling comparison.
• Construction of an auxiliary problem on $ \Omega^* $: $ -\Delta h = h^q $ in $ \Omega^* $, $ h = 0 $ on $ \partial\Omega^* $, to serve as a comparison solution.
• Application of the rearrangement method to compare the decreasing rearrangements of $ u $ and $ h $, leveraging known sharp inequalities from Chiti’s work.
• Derivation of bounds via normalization and scaling: the $ L^k $-norm and sup-norm of $ u $ are controlled by those of $ h $, scaled by the $ L^{q+1} $-norm of $ u $.
• Use of Faber-Krahn type inequalities and Green’s function estimates to bound the maximum of $ h $, leading to explicit dependence on domain volume.
• Employment of the functional $ S_q(\Omega) = \|u\|_{L^{q+1}(\Omega)}^{q-1} $ to relate norms across domains and establish monotonicity under symmetrization.

Experimental results

Research questions

• RQ1 What is the sharp upper bound for the $ L^k $-norm of the positive solution to $ -\Delta u = u^q $ in a bounded domain $ \Omega $, in terms of its $ L^{q+1} $-norm?
• RQ2 Can the maximum pointwise value of the solution be bounded from above by a functional of its $ L^{q+1} $-norm, with equality if and only if $ \Omega $ is a ball?
• RQ3 How do isoperimetric inequalities for solutions of sublinear elliptic equations compare to those known for eigenfunctions of the Laplacian?
• RQ4 What is the role of Schwarz symmetrization and comparison with a radial solution on a ball in deriving sharp estimates for nonlinear problems?
• RQ5 Can the sharpness of the estimates be characterized by domain symmetry, and under what conditions is equality achieved?

Key findings

• The $ L^k $-norm of the solution $ u $ satisfies $ \int_\Omega u^k dx \leq C(q,k,\Omega^*) \|u\|_{L^{q+1}(\Omega)}^{\sigma_1} $, where $ \sigma_1 = \frac{2(1+q)k + (1-q^2)n}{n+2 - (n-2)q} $, with equality if and only if $ \Omega $ is a ball.
• The essential supremum of $ u $ is bounded by $ \max_x u(x) \leq C(q,\Omega^*) \|u\|_{L^{q+1}(\Omega)}^{\sigma_2} $, where $ \sigma_2 = \frac{2(1+q)}{n+2 - (n-2)q} $, and equality holds if and only if $ \Omega $ is a ball.
• The constant $ C(q,\Omega^*) $ is defined as $ \max_{x\in\Omega^*} h(x) / \|h\|_{L^{q+1}(\Omega^*)}^{\sigma_2} $, where $ h $ solves the symmetrized problem on $ \Omega^* $, and is explicitly computable via Green’s function estimates.
• The maximum of $ h $ on $ \Omega^* $ is bounded by $ \left[ \frac{ |\Omega| }{ \omega_n (2n)^{n/2} } \right]^{2/((1-q)n)} $, leading to a volume-dependent sharp upper bound for $ \|u\|_{L^\infty} $.
• The $ L^k $-norm of $ u $ is dominated by that of $ h $, i.e., $ \int_\Omega u^k dx \leq \int_{\Omega^*} h^k dx $, under the assumption $ k \geq q+1 $, with equality iff $ \Omega $ is a ball.
• The sharpness of the estimates is proven via the equality condition in the symmetrization comparison, which holds only when $ \Omega $ is a ball, confirming the extremality of radial domains.