Modulus of continuity of solutions to complex Hessian equations (Q2790322): Difference between revisions
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scientific article | scientific article; zbMATH DE number 6549422 | ||
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It is known that the Dirichlet problem for the complex Hessian equation in a strongly \(m\)-pseudoconvex domain in \(\mathbb C ^n\), with \(C^{1,1}\) boundary data and the density in \(L^p\), \(p>n/m\), has a continuous solution. The bound for \(p\) is optimal. In a recent paper, \textit{N. C. Nguyen} [Potential Anal. 41, No. 3, 887--902 (2014; Zbl 1302.32034)] proved Hölder continuity of those solutions with an explicit exponent, but assuming a controlled growth of the right hand side near the boundary. In the present paper the author is able to remove this extra assumption. For \(p>2m/n\) he obtains also a better Hölder exponent. On the way, a sharp result on the modulus of continuity of the solution, when the right hand side is continuous, is shown. The proof uses a rather delicate barrier construction. | |||
| Property / review text: It is known that the Dirichlet problem for the complex Hessian equation in a strongly \(m\)-pseudoconvex domain in \(\mathbb C ^n\), with \(C^{1,1}\) boundary data and the density in \(L^p\), \(p>n/m\), has a continuous solution. The bound for \(p\) is optimal. In a recent paper, \textit{N. C. Nguyen} [Potential Anal. 41, No. 3, 887--902 (2014; Zbl 1302.32034)] proved Hölder continuity of those solutions with an explicit exponent, but assuming a controlled growth of the right hand side near the boundary. In the present paper the author is able to remove this extra assumption. For \(p>2m/n\) he obtains also a better Hölder exponent. On the way, a sharp result on the modulus of continuity of the solution, when the right hand side is continuous, is shown. The proof uses a rather delicate barrier construction. / rank | |||
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| Property / reviewed by | |||
| Property / reviewed by: Slawomir Kołodziej / rank | |||
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Latest revision as of 15:58, 22 May 2025
scientific article; zbMATH DE number 6549422
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Modulus of continuity of solutions to complex Hessian equations |
scientific article; zbMATH DE number 6549422 |
Statements
Modulus of continuity of solutions to complex Hessian equations (English)
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3 March 2016
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complex Hessian equations
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Hölder continuity
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modulus of continuity
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It is known that the Dirichlet problem for the complex Hessian equation in a strongly \(m\)-pseudoconvex domain in \(\mathbb C ^n\), with \(C^{1,1}\) boundary data and the density in \(L^p\), \(p>n/m\), has a continuous solution. The bound for \(p\) is optimal. In a recent paper, \textit{N. C. Nguyen} [Potential Anal. 41, No. 3, 887--902 (2014; Zbl 1302.32034)] proved Hölder continuity of those solutions with an explicit exponent, but assuming a controlled growth of the right hand side near the boundary. In the present paper the author is able to remove this extra assumption. For \(p>2m/n\) he obtains also a better Hölder exponent. On the way, a sharp result on the modulus of continuity of the solution, when the right hand side is continuous, is shown. The proof uses a rather delicate barrier construction.
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