The heat equation under conditions on the moments in higher dimensions. Issue 2 (23rd September 2014)

• Record Type: Journal Article
• Authors: Mugnolo, Delio; Nicaise, Serge
• Is Part Of: Mathematische Nachrichten. Volume 288:Issue 2/3(2015)
• Page Start: 295
• Page End: 308
• Publication Date: 2014-09-23
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201300298
• Languages: English
• ISSNs: 0025-584X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>We consider the heat equation on the <italic>N</italic>‐dimensional cube (0, 1)<sup><italic>N</italic></sup> and impose different classes of integral conditions, instead of usual boundary ones. Well‐posedness results for the heat equation under the condition that the moments of order 0 and 1 are conserved had been known so far only in the case of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh3hv4cfj1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:dummy:media:mana201300298:mana201300298-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>, for which such conditions can be easily interpreted as conservation of mass and barycenter. In this paper we show that in the case of general <italic>N</italic> the heat equation with such integral conditions is still well‐posed, upon suitably relaxing the notion of solution. Existence of solutions with general initial data in a suitable space of distributions over (0, 1)<sup><italic>N</italic></sup> are proved by introducing two appropriate realizations of the Laplacian and checking by form methods that they generate analytic semigroups. The solution thus obtained turns out to solve the heat equation only in a certain distributional sense. However, one of these realizations is tightly related to a well‐known object of operator theory, the Krein–von Neumann extension of the Laplacian. This connection also establishes well‐posedness in a classical sense, as long as the initial data are <italic>L</italic><sup>2</sup>‐functions.</p> </abstract>