The Monge–Ampère System: Convex Integration in Arbitrary Dimension and Codimension
Marta Lewicka
Source abstract
Abstract. In this paper, we study flexibility of the weak solutions to the Monge–Ampère system (MA) via convex integration. This new system of PDEs is an extension of the Monge–Ampère equation in [Formula: see text] dimensions, naturally arising from the prescribed curvature problem and closely related to the classical problem of the isometric immersions (II). Our main result achieves density in the set of subsolutions, of the Hölder [Formula: see text] solutions to the Von Kármán system (VK) which is the weak formulation of (MA). The regularity exponent [Formula: see text] is any exponent satisfying [Formula: see text], where [Formula: see text] is an arbitrary dimension and [Formula: see text] an arbitrary codimension of the problem. At [Formula: see text], this agrees with flexibility up to [Formula: see text] for (II) with [Formula: see text] proved by Conti and Szekelyhidi in [ Sci. China Math., 62 (2019), pp. 1041–1056 ]. At [Formula: see text], [Formula: see text], this extends the initial findings by the author and Pakzad in [ Anal. PDE, 10 (2017), pp. 695–727 ] for (MA). Our result covers the codimension interval [Formula: see text] so far uncharted even for the system (II), since the regularity [Formula: see text] with any [Formula: see text] achieved by Källen in [ Ark. Mat, 16 (1978), pp. 29–50 ], strictly requires a large codimension. Our second main result reproduces Källen’s result in the context of (MA), obtaining density in the set of subsolutions, of [Formula: see text] regular solutions for any [Formula: see text] whenever [Formula: see text]. As an application of our results for (VK), we derive an energy scaling bound in the quantitative immersability of Riemannian metrics for nonlinear energy functionals modeled on the energies of deformations of thin prestrained films in the nonlinear elasticity [ M. Lewicka, Calculus of Variations on Thin Prestressed Films: Asymptotic Methods in Elasticity, Progr. Nonlinear Differential Equations Appl. 101, Birkhäuser, Boston, 2022 ].
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