[Paper Review] On the maximum principle for parabolic equations with unbounded coefficients

Alexander I. Nazarov|arXiv (Cornell University)|Jul 18, 2015

Differential Equations and Boundary Problems6 references3 citations

TL;DR

This paper establishes sharp maximum principle estimates for parabolic equations with unbounded coefficients, extending prior results by allowing coefficients in Lorentz-type Lebesgue spaces $ L_p^xL_q^t $ with $ \frac{n}{p} + \frac{1}{q} \leq 1 $. It introduces a novel pivotal lemma and proves estimates for 'composite' coefficients $ b_i = \sum b_i^{(k)} $, where each $ b_i^{(k)} $ belongs to such a space, achieving optimal integrability conditions via weighted norms and Bony-type maximum principles.

ABSTRACT

Translation of the paper "Interpolation of linear spaces and maximum estimates for solutions to parabolic equations" published in Russian in the collected volume "Partial differential equations", Akad. Nauk SSSR, Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1987, 50--72. Original proofs are essentially simplified. Some gaps are fixed and some comments are added.

Motivation & Objective

• To extend the maximum principle estimates for parabolic equations to the case of unbounded coefficients, particularly in Lorentz-type Lebesgue spaces $ L_p^xL_q^t $ with $ \frac{n}{p} + \frac{1}{q} \leq 1 $.
• To generalize existing results by allowing coefficients $ b_i $ to be in $ L_p^xL_q^t $ or in a 'composite' form $ b_i = \sum b_i^{(k)} $ with $ b_i^{(k)} \in L_{p_k}^xL_{q_k}^t $.
• To prove a Bony-type maximum principle for parabolic operators under these generalized coefficient conditions.
• To unify and simplify earlier proofs by avoiding high-level arguments and fixing gaps in prior works.

Proposed method

• Introduces a pivotal lemma that provides a priori estimates for solutions in terms of the right-hand side $ f $, using weighted norms involving $ \sigma $, $ \det(a) $, and $ c $.
• Applies Minkowski’s inequality to relate the mixed-norm spaces $ L_p^xL_q^t $ and $ L_q^tL_p^x $, ensuring continuity of embeddings under $ p \leq q $ or $ p \geq q $.
• Uses a transformation via $ v $-functions satisfying a Monge-Ampère-type equation to construct supersolutions $ B $ satisfying $ \mathcal{L}B \geq |b| $, enabling the application of the maximum principle.
• For composite coefficients, decomposes $ b_i $ into components $ b_i^{(k)} $, constructs separate supersolutions $ B_k $, and combines them with a normalization factor involving $ \|DB_1\|_{C_{RT}} $ to control the total growth.
• Employs the method of sub- and supersolutions combined with $ L_p $-norm estimates for $ b_i^{(k)} $, using the weight $ h_k = \sigma^{-1/q_k}(\det a)^{-1/p_k} c^{n/p_k + 1/q_k - 1} |b^{(k)}| $.
• Applies the Bony-type maximum principle by contradiction: assuming $ \mathcal{L}u < 0 $ at an interior maximum leads to a contradiction via decay estimates in $ L_{p_k,q_k} $-norms.

Experimental results

Research questions

• RQ1 Can the maximum principle estimate for parabolic equations be extended to unbounded coefficients in mixed-norm Lebesgue spaces $ L_p^xL_q^t $ with $ \frac{n}{p} + \frac{1}{q} \leq 1 $?
• RQ2 How can the estimate be generalized when the lower-order coefficient $ b_i $ is not in a single $ L_p^xL_q^t $ space but is a sum of components in different such spaces?
• RQ3 What is the optimal integrability condition on $ b_i $ that still allows for a priori $ L^\infty $-bound on the solution via the maximum principle?
• RQ4 Can the Bony-type maximum principle be extended to parabolic operators with unbounded coefficients in Lorentz-type spaces?
• RQ5 What modifications are needed in the proof strategy to handle composite coefficients without relying on boundedness assumptions?

Key findings

• The paper establishes the estimate $ \|u\|_{L^\infty(Q)} \leq N \left\| \frac{(\mathcal{L}u)_+}{(\sigma \det a)^{1/(n+1)}} \right\|_{n+1,Q} $ for solutions in $ W^{2,1}_{n+1}(Q) \cap C(\overline{Q}) $, under non-degenerate assumptions.
• For coefficients $ b_i \in L_p^xL_q^t(Q) $ with $ \frac{n}{p} + \frac{1}{q} \leq 1 $, the maximum principle estimate holds with a constant depending on $ n $, $ R $, and the norm of $ b_i $ in the respective space.
• The estimate is extended to 'composite' coefficients $ b_i = \sum_{k=1}^m b_i^{(k)} $, where each $ b_i^{(k)} \in L_{p_k}^xL_{q_k}^t(Q) $, provided $ \frac{n}{p_k} + \frac{1}{q_k} \leq 1 $, with the constant depending on $ \|h_k\|_{p_k,q_k,Q} $, where $ h_k = \sigma^{-1/q_k}(\det a)^{-1/p_k} c^{n/p_k + 1/q_k - 1} |b^{(k)}| $.
• The Bony-type maximum principle is proven: if a solution $ u \in W^{2,1}_{p_0,q_0,\text{loc}}(Q) $ attains a non-negative maximum in the interior of $ Q $, then $ \sup_Q \frac{\mathcal{L}u}{{\bf Sp}(a)+\sigma+c} \geq 0 $, under the same integrability assumptions.
• The proof technique avoids reliance on high-level tools from [BL] and [KP], simplifying earlier arguments in [N87] and [N88], and fixes gaps in auxiliary assertions.
• The result is optimal in the sense that the condition $ \frac{n}{p} + \frac{1}{q} \leq 1 $ cannot be relaxed, as shown by counterexamples in the literature and the structure of the weight functions used.