Document Zbl 1531.35167

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Math.</a> <a href="/?q=in%3A507215" title="Articles in this Issue">79, No. 2, Paper No. 53, 22 p. (2024)</a>. </div> <div class="abstract">Summary: In this paper, we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in $\mathbb{R}^{n+1}$ with speed $\psi \sigma_k^{\alpha}(\lambda)$, where $\alpha >\frac{1}{k}$ is a constant, $\sigma_k(\lambda)$ is the $k$-th elementary symmetric polynomial of the principal radii of curvature and $\psi$ is a preassigned positive smooth function defined on $\mathbb{S}^n$. We prove that under some assumptions of $\psi$, the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known $L^p$ Christoffel-Minkowski problem $u^{1-p}(x) \sigma_k (\nabla^2u+uI)=c\psi (x)$ for $1<p<k+1$.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K93" title="MSC2020">35K93</a> </td> <td class="space"> Quasilinear parabolic equations with mean curvature operator </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35B45" title="MSC2020">35B45</a> </td> <td class="space"> A priori estimates in context of PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A52A20" title="MSC2020">52A20</a> </td> <td class="space"> Convex sets in $n$ dimensions (including convex hypersurfaces) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acurvature+flows">curvature flows</a>; <a href="/?q=ut%3AChristoffel-Minkowski+problem">Christoffel-Minkowski problem</a>; <a href="/?q=ut%3Agradient+estimates">gradient estimates</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1531.35167" data-ciurl="/ci/07791783" data-biburl="/bibtex/07791783.bib" data-amsurl="/amsrefs/07791783.bib" data-xmlurl="/xml/07791783.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07791783.pdf" title="Zbl 1531.35167 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00025-023-02069-0" aria-label="DOI for “A curvature flow approach to $L^p$ Christoffel-Minkowski problem for $1<p<k+1$”" title="10.1007/s00025-023-02069-0">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2301.06491"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Alexandroff, A.: Über die Oberflächenfunktion eines konvexen Körpers. (Bemerkung zur Arbeit Zur Theorie der gemischten Volumina von konvexen Körpern”), Rec. Math. N.S. [Mat. Sbornik], 6(48), 167-174 (1939) · <a href="/0022.40203" class="nowrap">Zbl 0022.40203</a></td> </tr><tr> <td>[2]</td> <td class="space">Alexandroff, A., Smoothness of the convex surface of bounded Gaussian curvature. In: C. R. (Doklady) Acad. Sci. URSS (N.S.), 36, 195-199 (1942) · <a href="/0061.37605" class="nowrap">Zbl 0061.37605</a></td> </tr><tr> <td>[3]</td> <td class="space">Aleksandrov, A.D.: Dirichlet’s problem for the equation ${\rm Det}\,||z_{ij}|| =\varphi (z_1,\cdots ,z_n,z, x_1,\cdots , x_n)$. I, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astr., 13, 5-24 (1958) · <a href="/0114.30202" class="nowrap">Zbl 0114.30202</a></td> </tr><tr> <td>[4]</td> <td class="space">Andrews, Ben, Evolving convex curves, Calc. Var. Partial Differ. Equ., 7, 4, 315-371 (1998) · <a href="/0931.53030" class="nowrap">Zbl 0931.53030</a> · <a href="https://doi.org/10.1007/s005260050111" class="nowrap">doi:10.1007/s005260050111</a></td> </tr><tr> <td>[5]</td> <td class="space">Andrews, Ben, Pinching estimates and motion of hypersurfaces by curvature functions, J. Reine Angew. Math., 608, 17-33 (2007) · <a href="/1129.53044" class="nowrap">Zbl 1129.53044</a></td> </tr><tr> <td>[6]</td> <td class="space">Bryan, P.; Ivaki, M.; Scheuer, J., Christoffel-Minkowski flows, Trans. Am. Math. Soc., 376, 4, 2373-2393 (2023) · <a href="/1510.35133" class="nowrap">Zbl 1510.35133</a></td> </tr><tr> <td>[7]</td> <td class="space">Calabi, E., Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J., 5, 105-126 (1958) · <a href="/0113.30104" class="nowrap">Zbl 0113.30104</a> · <a href="https://doi.org/10.1307/mmj/1028998055" class="nowrap">doi:10.1307/mmj/1028998055</a></td> </tr><tr> <td>[8]</td> <td class="space">Cheng, Shiu Yuen; Yau, Shing Tung, On the regularity of the solution of the $n$-dimensional Minkowski problem, Comm. Pure Appl. Math., 29, 5, 495-516 (1976) · <a href="/0363.53030" class="nowrap">Zbl 0363.53030</a> · <a href="https://doi.org/10.1002/cpa.3160290504" class="nowrap">doi:10.1002/cpa.3160290504</a></td> </tr><tr> <td>[9]</td> <td class="space">Chou, KS; Wang, XJ, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math., 205, 1, 33-83 (2006) · <a href="/1245.52001" class="nowrap">Zbl 1245.52001</a> · <a href="https://doi.org/10.1016/j.aim.2005.07.004" class="nowrap">doi:10.1016/j.aim.2005.07.004</a></td> </tr><tr> <td>[10]</td> <td class="space">Chow, B.; Gulliver, R., Aleksandrov reflection and nonlinear evolution equations. I. The $n$-sphere and $n$-ball, Calc. Var. Partial Differ. Equ., 4, 3, 249-264 (1996) · <a href="/0851.58041" class="nowrap">Zbl 0851.58041</a> · <a href="https://doi.org/10.1007/BF01254346" class="nowrap">doi:10.1007/BF01254346</a></td> </tr><tr> <td>[11]</td> <td class="space">Ding, S.; Li, G., A class of inverse curvature flows and $L^p$ dual Christoffel-Minkowski problem, Trans. Amer. Math. Soc., 376, 1, 697-752 (2023) · <a href="/1512.53089" class="nowrap">Zbl 1512.53089</a></td> </tr><tr> <td>[12]</td> <td class="space">Firey, WJ, $p$-means of convex bodies, Math. Scand., 10, 17-24 (1962) · <a href="/0188.27303" class="nowrap">Zbl 0188.27303</a> · <a href="https://doi.org/10.7146/math.scand.a-10510" class="nowrap">doi:10.7146/math.scand.a-10510</a></td> </tr><tr> <td>[13]</td> <td class="space">Gerhardt, C., Flow of nonconvex hypersurfaces into spheres, J. Differ. Geom., 32, 1, 299-314 (1990) · <a href="/0708.53045" class="nowrap">Zbl 0708.53045</a> · <a href="https://doi.org/10.4310/jdg/1214445048" class="nowrap">doi:10.4310/jdg/1214445048</a></td> </tr><tr> <td>[14]</td> <td class="space">Gerhardt, Claus, Non-scale-invariant inverse curvature flows in Euclidean space, Calc. Var. Partial Differ. Equ., 49, 1-2, 471-489 (2014) · <a href="/1284.53062" class="nowrap">Zbl 1284.53062</a> · <a href="https://doi.org/10.1007/s00526-012-0589-x" class="nowrap">doi:10.1007/s00526-012-0589-x</a></td> </tr><tr> <td>[15]</td> <td class="space">Guan, P.; Ma, XN, The Christoffel-Minkowski problem. I. Convexity of solutions of a Hessian equation, Invent. Math., 151, 3, 553-577 (2003) · <a href="/1213.35213" class="nowrap">Zbl 1213.35213</a> · <a href="https://doi.org/10.1007/s00222-002-0259-2" class="nowrap">doi:10.1007/s00222-002-0259-2</a></td> </tr><tr> <td>[16]</td> <td class="space">Guan, P.; Ma, XN; Trudinger, N.; Zhu, X., A form of Alexandrov-Fenchel inequality, Pure Appl. Math. Q., 6, 4, 999-1012 (2010) · <a href="/1230.52020" class="nowrap">Zbl 1230.52020</a> · <a href="https://doi.org/10.4310/PAMQ.2010.v6.n4.a2" class="nowrap">doi:10.4310/PAMQ.2010.v6.n4.a2</a></td> </tr><tr> <td>[17]</td> <td class="space">Guan, P.; Xia, C., $L^p$ Christoffel-Minkowski problem: the case $1<p<k+1$, Calc. Var. Partial Differ. Equ., 57, 2, 23 (2018) · <a href="/1395.52005" class="nowrap">Zbl 1395.52005</a> · <a href="https://doi.org/10.1007/s00526-018-1341-y" class="nowrap">doi:10.1007/s00526-018-1341-y</a></td> </tr><tr> <td>[18]</td> <td class="space">Heinz, Erhard, On elliptic Monge-Ampère equations and Weyl’s embedding problem, J. Analyse Math., 7, 1-52 (1959) · <a href="/0152.30901" class="nowrap">Zbl 0152.30901</a> · <a href="https://doi.org/10.1007/BF02787679" class="nowrap">doi:10.1007/BF02787679</a></td> </tr><tr> <td>[19]</td> <td class="space">Hu, C.; Ma, XN; Shen, C., On the Christoffel-Minkowski problem of Firey’s $p$-sum, Calc. Var. Partial Differ. Equ., 21, 2, 137-155 (2004) · <a href="/1161.35391" class="nowrap">Zbl 1161.35391</a> · <a href="https://doi.org/10.1007/s00526-003-0250-9" class="nowrap">doi:10.1007/s00526-003-0250-9</a></td> </tr><tr> <td>[20]</td> <td class="space">Ivaki, Mohammad N., Deforming a hypersurface by principal radii of curvature and support function, Calc. Var. Partial Differ. Equ., 58, 1, 18 (2019) · <a href="/1403.53057" class="nowrap">Zbl 1403.53057</a> · <a href="https://doi.org/10.1007/s00526-018-1462-3" class="nowrap">doi:10.1007/s00526-018-1462-3</a></td> </tr><tr> <td>[21]</td> <td class="space">Krylov, NV, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat., 46, 3, 487-523 (1982) · <a href="/0511.35002" class="nowrap">Zbl 0511.35002</a></td> </tr><tr> <td>[22]</td> <td class="space">Lewy, H., On differential geometry in the large. I. Minkowski’s problem, Trans. Amer. Math. Soc., 43, 2, 258-270 (1938) · <a href="/64.0714.03" class="nowrap">JFM 64.0714.03</a></td> </tr><tr> <td>[23]</td> <td class="space">Li, QR; Sheng, W.; Wang, XJ, Asymptotic convergence for a class of fully nonlinear curvature flows, J. Geom. Anal., 30, 1, 834-860 (2020) · <a href="/1434.53096" class="nowrap">Zbl 1434.53096</a> · <a href="https://doi.org/10.1007/s12220-019-00169-4" class="nowrap">doi:10.1007/s12220-019-00169-4</a></td> </tr><tr> <td>[24]</td> <td class="space">Lieberman, G.M.: Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, xii+439 (1996) · <a href="/0884.35001" class="nowrap">Zbl 0884.35001</a></td> </tr><tr> <td>[25]</td> <td class="space">Lu, J.; Wang, XJ, Rotationally symmetric solutions to the $L_p$-Minkowski problem, J. Differ. Equ., 254, 3, 983-1005 (2013) · <a href="/1273.52006" class="nowrap">Zbl 1273.52006</a> · <a href="https://doi.org/10.1016/j.jde.2012.10.008" class="nowrap">doi:10.1016/j.jde.2012.10.008</a></td> </tr><tr> <td>[26]</td> <td class="space">Lutwak, E., The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differ. Geom., 38, 1, 131-150 (1993) · <a href="/0788.52007" class="nowrap">Zbl 0788.52007</a> · <a href="https://doi.org/10.4310/jdg/1214454097" class="nowrap">doi:10.4310/jdg/1214454097</a></td> </tr><tr> <td>[27]</td> <td class="space">Lutwak, E.; Oliker, V., On the regularity of solutions to a generalization of the Minkowski problem, J. Differ. Geom., 41, 1, 227-246 (1995) · <a href="/0867.52003" class="nowrap">Zbl 0867.52003</a> · <a href="https://doi.org/10.4310/jdg/1214456011" class="nowrap">doi:10.4310/jdg/1214456011</a></td> </tr><tr> <td>[28]</td> <td class="space">Minkowski, H., Volumen und Oberfläche, Math. Ann., 57, 4, 447-495 (1903) · <a href="/34.0649.01" class="nowrap">JFM 34.0649.01</a> · <a href="https://doi.org/10.1007/BF01445180" class="nowrap">doi:10.1007/BF01445180</a></td> </tr><tr> <td>[29]</td> <td class="space">Nirenberg, Louis, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., 6, 337-394 (1953) · <a href="/0051.12402" class="nowrap">Zbl 0051.12402</a> · <a href="https://doi.org/10.1002/cpa.3160060303" class="nowrap">doi:10.1002/cpa.3160060303</a></td> </tr><tr> <td>[30]</td> <td class="space">Pogorelov, AV, Regularity of a convex surface with given Gaussian curvature, Mat. Sbornik N.S., 31, 73, 88-103 (1952) · <a href="/0048.40501" class="nowrap">Zbl 0048.40501</a></td> </tr><tr> <td>[31]</td> <td class="space">Pogorelov, AV, A regular solution of the $n$-dimensional Minkowski problem, Soviet Math. Dokl., 12, 1192-1196 (1971) · <a href="/0236.53057" class="nowrap">Zbl 0236.53057</a></td> </tr><tr> <td>[32]</td> <td class="space">Sheng, W.; Yi, C., A class of anisotropic expanding curvature flows, Discrete Contin. Dyn. Syst., 40, 4, 2017-2035 (2020) · <a href="/1433.53118" class="nowrap">Zbl 1433.53118</a> · <a href="https://doi.org/10.3934/dcds.2020104" class="nowrap">doi:10.3934/dcds.2020104</a></td> </tr><tr> <td>[33]</td> <td class="space">Urbas, JIE, An expansion of convex hypersurfaces, J. Differ. Geom., 33, 1, 91-125 (1991) · <a href="/0746.53006" class="nowrap">Zbl 0746.53006</a> · <a href="https://doi.org/10.4310/jdg/1214446031" class="nowrap">doi:10.4310/jdg/1214446031</a></td> </tr><tr> <td>[34]</td> <td class="space">Wang, X.J.: The $k$-Hessian equation, Geometric analysis and PDEs, Lecture Notes in Math. Springer. Dordrecht 2009, 177-252 (1977) · <a href="/1196.35093" class="nowrap">Zbl 1196.35093</a></td> </tr><tr> <td>[35]</td> <td class="space">Zhu, G., The $L_p$ Minkowski problem for polytopes for $0<p<1$, J. Funct. Anal., 269, 4, 1070-1094 (2015) · <a href="/1335.52023" class="nowrap">Zbl 1335.52023</a> · <a href="https://doi.org/10.1016/j.jfa.2015.05.007" class="nowrap">doi:10.1016/j.jfa.2015.05.007</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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