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The distributional \(k\)-Hessian in fractional Sobolev spaces. (English) Zbl 1454.46035

Summary: We introduce the notion of a distributional \(k\)-Hessian \((k = 2, \ldots, n)\) associated with fractional Sobolev functions on \(\Omega\), a smooth bounded open subset in \(\mathbb{R}^n\). We show that the distributional \(k\)-Hessian is weakly continuous on the fractional Sobolev space \(W^{2 - 2/k, k} (\Omega)\) and that the weak continuity result is optimal, that is, the distributional \(k\)-Hessian is well defined in \(W^{s, p}(\Omega)\) if and only if \(W^{s, p}(\Omega) \subseteq W^{2 - 2/k, k}(\Omega)\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J60 Nonlinear elliptic equations
46F10 Operations with distributions and generalized functions
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[1] Baer, E. and Jerison, D., ‘Optimal function spaces for continuity of the Hessian determinant as a distribution’, J. Funct. Anal.269 (2015), 1482-1514. · Zbl 1335.46027
[2] Brezis, H. and Nguyen, H., ‘The Jacobian determinant revisited’, Invent. Math.185 (2011), 17-54. · Zbl 1230.46029
[3] Colesanti, A. and Hug, D., ‘Hessian measures of semi-convex functions and applications to support measures of convex bodies’, Manuscripta Math.101 (2000), 209-238. · Zbl 0973.52003
[4] Colesanti, A. and Salani, P., ‘Generalized solutions of Hessian equations’, Bull. Aust. Math. Soc.56 (1997), 459-466. · Zbl 0895.35033
[5] Fu, J., ‘Monge-Ampère functions I’, Indiana Univ. Math. J.38 (1989), 745-771. · Zbl 0668.49010
[6] Giaquinta, M., Modica, G. and Souček, J., Cartesian Currents in the Calculus of Variations, I (Springer, Berlin, 1998). · Zbl 0914.49001
[7] Iwaniec, T., ‘On the concept of the weak Jacobian and Hessian’, Papers on analysis: A volume dedicated to Olli Martio on the occasion of his 60th birthday, Rep. Univ. Jyväskylä Dep. Math. Stat.83 (2001), 181-205. · Zbl 1007.35015
[8] Stein, E., ‘The characterization of functions arising as potentials. I’, Bull. Amer. Math. Soc. (N.S.)67 (1961), 102-104. · Zbl 0127.32002
[9] Stein, E., ‘The characterization of functions arising as potentials. II’, Bull. Amer. Math. Soc. (N.S.)68 (1962), 577-582.
[10] Strichartz, R., ‘Fubini-type theorems’, Ann. Scuola Norm. Sup. Pisa (3)22 (1968), 399-408. · Zbl 0165.14602
[11] Treibel, H., Theory of Functions Spaces, (Birkhauser Verlag, Basel, 1983). · Zbl 0546.46027
[12] Trudinger, N. and Wang, X., ‘Hessian measures I’, Topol. Methods Nonlinear Anal.10 (1997), 225-239. · Zbl 0915.35039
[13] Trudinger, N. and Wang, X., ‘Hessian measures II’, Ann. of Math. (2)150 (1999), 579-604. · Zbl 0947.35055
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