An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability

An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence,... References[1] L. Ambrosio, Fine properties of sets of finite perimeter in doublingmetric measure spaces, Calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10 (2002), no. 2-3, 111-128.[2] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems., Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.[3] L. Ambrosio, M. Miranda, Jr., and D. Pallara, Special functions of bounded variation in doubling metric measure spaces, Calculus of variations: topics from the mathematical heritage of E. De Giorgi, 1-45, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.[4] A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17. European Mathematical Society (EMS), Zürich, 2011. xii+403 pp.[5] A. Björn and J. Björn, Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology, Rev. Mat. Iberoam. 31 (2015), no. 1, 161-214.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000352568500007&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[6] A. Björn, J. Björn, and J. Malý, Quasiopen and p-path open sets, and characterizations of quasicontinuity, Potential Anal. 46 (2017), no. 1, 181-199. doi: 10.1007/s11118-016-9580-z.10.1007/s11118-016-9580-z[7] A. Björn, J. Björn, and N. Shanmugalingam, Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces, Houston J. Math. 34 (2008), no. 4, 1197-1211.[8] A. Björn, J. Björn, and N. Shanmugalingam, The Dirichlet problemfor p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173-203.[9] A. Björn, J. Björn, and N. Shanmugalingam, The Dirichlet problemfor p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities, J. Differential Equations 259 (2015), no. 7, 3078-3114.[10] E. Durand-Cartagena, and A. Lemenant, Some stability results under domain variation for Neumann problems in metric spaces, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 537-563.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000282496400012&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[11] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in AdvancedMathematics series, CRC Press, Boca Raton, 1992.[12] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp.[13] B. Fuglede, The quasi topology associated with a countably subadditive set function, Ann. Inst. Fourier 21 (1971), no. 1, 123-169.[14] E. Giusti,Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. xii+240 pp.[15] P. Hajłasz, Sobolev spaces on metric-measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173-218. Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.[16] H. Hakkarainen and J. Kinnunen, The BV-capacity in metric spaces, Manuscripta Math. 132 (2010), no. 1-2, 51-73.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000276432900003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[17] H. Hakkarainen, J. Kinnunen, P. Lahti, and P. Lehtelä, Relaxation and integral representation for functionals of linear growth on metric measures spaces, Anal. Geom. Metr. Spaces 4 (2016), 288-313.[18] H. Hakkarainen, R. Korte, P. Lahti, and N. Shanmugalingam, Stability and continuity of functions of least gradient, Anal. Geom. Metr. Spaces 3 (2015), 123-139.[19] H. Hakkarainen and N. Shanmugalingam, Comparisons of relative BV-capacities and Sobolev capacity in metric spaces, Nonlinear Anal. 74 (2011), no. 16, 5525-5543.[20] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1-61.[21] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. Tyson, Sobolev spaces on metric measure spaces. An approach based on upper gradients, New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015. xii+434 pp.[22] J. Kinnunen, R. Korte, A. Lorent, and N. Shanmugalingam, Regularity of sets with quasiminimal boundary surfaces in metric spaces, J. Geom. Anal. 23 (2013), no. 4, 1607-1640.10.1007/s12220-012-9299-zhttp://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000325065700001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[23] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, A characterization of Newtonian functions with zero boundary values, Calc. Var. Partial Differential Equations 43 (2012), no. 3-4, 507-528.[24] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, Lebesgue points and capacities via the boxing inequality in metric spaces, Indiana Univ. Math. J. 57 (2008), no. 1, 401-430.[25] R. Korte, P. Lahti, X. Li, and N. Shanmugalingam, Notions of Dirichlet problemfor functions of least gradient inmetricmeasure spaces, preprint 2016. http://cvgmt.sns.it/paper/3295/[26] P. Lahti, Extensions and traces of functions of bounded variation on metric spaces, J. Math. Anal. Appl. 423 (2015), no. 1, 521-537.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000349706000031&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[27] P. Lahti, Quasiopen sets, bounded variation and lower semicontinuity in metric spaces, preprint 2017. arXiv:1703.04675[28] P. Lahti, Strong approximation of sets of finite perimeter in metric spaces, to appear in manuscripta mathematica. doi: 10.1007/s00229-017-0948-110.1007/s00229-017-0948-1[29] P. Lahti, L. Malý, N. Shanmugalingam, and G. Speight, Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient, preprint 2017. http://cvgmt.sns.it/paper/3500/[30] P. Lahti and N. Shanmugalingam, Fine properties and a notion of quasicontinuity for BV functions on metric spaces, J. Math. Pures Appl. (9) 107 (2017), no. 2, 150-182.[31] P. Lahti and N. Shanmugalingam, Trace theorems for functions of bounded variation in metric spaces, preprint 2015. arXiv:1507.07006[32] L. Malý and N. Shanmugalingam, Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity, preprint 2016. arXiv:1609.06808[33] L. Malý, N. Shanmugalingam, and M. Snipes, Trace and extension theorems for functions of bounded variation, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. doi: 10.2422/2036-2145.201511_00710.2422/2036-2145.201511_007[34] N. Marola, M. Miranda Jr., and N. Shanmugalingam, Boundary measures, generalized Gauss-Green formulas and the mean value property in metric measure spaces Revista Mat. Iberoamericana 31 (2015), no. 2, 497-530.[35] J. M. Mazón, J. D. Rossi, and S. Segura de León, Functions of least gradient and 1-harmonic functions. Indiana Univ. Math. J. 63 (2014), no. 4, 1067-1084.[36] J. M. Mazón, J. D. Rossi, and S. Segura de León, The 1-Laplacian elliptic equation with inhomogeneous Robin boundary conditions, Differential and Integral Equations 28 (2015) no. 5-6, 401-430.[37] A. Mercaldo, J. D. Rossi, S. Segura de León, and C. Trombetti, Behaviour of p-Laplacian problems with Neumann boundary conditions when p goes to 1, Commun. Pure Appl. Anal. 12 (2013), 253-267.[38] M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J.Math. Pures Appl. (9) 82 (2003), no. 8, 975-1004.[39] A. Moradifam, Least gradient problemswith Neumann boundary condition, J. Differential Equations 263 (2017), no. 11, 7900-7918.ams inequality on metric measure spaces, Rev. Mat. Iberoam. 25 (2009), no. 2, 533-558.[40] T. Mäkäläinen, Adams inequality on metric measure spaces, Rev. Mat. Iberoam. 25 (2009), no. 2, 533-558.10.4171/RMI/575[41] C. Scheven and T. Schmidt, On the dual formulation of obstacle problems for the total variation and the area functional, to appear in Ann. Inst. Henri Poincaré, Anal. Non Linéaire. http://cvgmt.sns.it/paper/3320/[42] N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), no. 3, 1021-1050.[43] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16(2) (2000), 243-279.[44] P. Sternberg, G. Williams, and W. Ziemer, Existence, uniqueness, and regularity for functions of least gradient, J. Reine Angew. Math. 430 (1992), 35-60.[45] W. P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989.[46] W. Ziemer and K. Zumbrun, The obstacle problemfor functions of least gradient,Math. Bohem. 124 (1999), no. 2-3, 193-219. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Geometry in Metric Spaces de Gruyter

An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability

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References[1] L. Ambrosio, Fine properties of sets of finite perimeter in doublingmetric measure spaces, Calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10 (2002), no. 2-3, 111-128.[2] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems., Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.[3] L. Ambrosio, M. Miranda, Jr., and D. Pallara, Special functions of bounded variation in doubling metric measure spaces, Calculus of variations: topics from the mathematical heritage of E. De Giorgi, 1-45, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.[4] A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17. European Mathematical Society (EMS), Zürich, 2011. xii+403 pp.[5] A. Björn and J. Björn, Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology, Rev. Mat. Iberoam. 31 (2015), no. 1, 161-214.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000352568500007&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[6] A. Björn, J. Björn, and J. Malý, Quasiopen and p-path open sets, and characterizations of quasicontinuity, Potential Anal. 46 (2017), no. 1, 181-199. doi: 10.1007/s11118-016-9580-z.10.1007/s11118-016-9580-z[7] A. Björn, J. Björn, and N. Shanmugalingam, Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces, Houston J. Math. 34 (2008), no. 4, 1197-1211.[8] A. Björn, J. Björn, and N. Shanmugalingam, The Dirichlet problemfor p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173-203.[9] A. Björn, J. Björn, and N. Shanmugalingam, The Dirichlet problemfor p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities, J. Differential Equations 259 (2015), no. 7, 3078-3114.[10] E. Durand-Cartagena, and A. Lemenant, Some stability results under domain variation for Neumann problems in metric spaces, Ann. Acad. Sci. Fenn. Math. 35 (2010), no. 2, 537-563.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000282496400012&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[11] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in AdvancedMathematics series, CRC Press, Boca Raton, 1992.[12] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969 xiv+676 pp.[13] B. Fuglede, The quasi topology associated with a countably subadditive set function, Ann. Inst. Fourier 21 (1971), no. 1, 123-169.[14] E. Giusti,Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. xii+240 pp.[15] P. Hajłasz, Sobolev spaces on metric-measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173-218. Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.[16] H. Hakkarainen and J. Kinnunen, The BV-capacity in metric spaces, Manuscripta Math. 132 (2010), no. 1-2, 51-73.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000276432900003&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[17] H. Hakkarainen, J. Kinnunen, P. Lahti, and P. Lehtelä, Relaxation and integral representation for functionals of linear growth on metric measures spaces, Anal. Geom. Metr. Spaces 4 (2016), 288-313.[18] H. Hakkarainen, R. Korte, P. Lahti, and N. Shanmugalingam, Stability and continuity of functions of least gradient, Anal. Geom. Metr. Spaces 3 (2015), 123-139.[19] H. Hakkarainen and N. Shanmugalingam, Comparisons of relative BV-capacities and Sobolev capacity in metric spaces, Nonlinear Anal. 74 (2011), no. 16, 5525-5543.[20] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1-61.[21] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. Tyson, Sobolev spaces on metric measure spaces. An approach based on upper gradients, New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015. xii+434 pp.[22] J. Kinnunen, R. Korte, A. Lorent, and N. Shanmugalingam, Regularity of sets with quasiminimal boundary surfaces in metric spaces, J. Geom. Anal. 23 (2013), no. 4, 1607-1640.10.1007/s12220-012-9299-zhttp://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000325065700001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[23] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, A characterization of Newtonian functions with zero boundary values, Calc. Var. Partial Differential Equations 43 (2012), no. 3-4, 507-528.[24] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, Lebesgue points and capacities via the boxing inequality in metric spaces, Indiana Univ. Math. J. 57 (2008), no. 1, 401-430.[25] R. Korte, P. Lahti, X. Li, and N. Shanmugalingam, Notions of Dirichlet problemfor functions of least gradient inmetricmeasure spaces, preprint 2016. http://cvgmt.sns.it/paper/3295/[26] P. Lahti, Extensions and traces of functions of bounded variation on metric spaces, J. Math. Anal. Appl. 423 (2015), no. 1, 521-537.http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000349706000031&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=b7bc2757938ac7a7a821505f8243d9f3[27] P. Lahti, Quasiopen sets, bounded variation and lower semicontinuity in metric spaces, preprint 2017. arXiv:1703.04675[28] P. Lahti, Strong approximation of sets of finite perimeter in metric spaces, to appear in manuscripta mathematica. doi: 10.1007/s00229-017-0948-110.1007/s00229-017-0948-1[29] P. Lahti, L. Malý, N. Shanmugalingam, and G. Speight, Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient, preprint 2017. http://cvgmt.sns.it/paper/3500/[30] P. Lahti and N. Shanmugalingam, Fine properties and a notion of quasicontinuity for BV functions on metric spaces, J. Math. Pures Appl. (9) 107 (2017), no. 2, 150-182.[31] P. Lahti and N. Shanmugalingam, Trace theorems for functions of bounded variation in metric spaces, preprint 2015. arXiv:1507.07006[32] L. Malý and N. Shanmugalingam, Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity, preprint 2016. arXiv:1609.06808[33] L. Malý, N. Shanmugalingam, and M. Snipes, Trace and extension theorems for functions of bounded variation, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. doi: 10.2422/2036-2145.201511_00710.2422/2036-2145.201511_007[34] N. Marola, M. Miranda Jr., and N. Shanmugalingam, Boundary measures, generalized Gauss-Green formulas and the mean value property in metric measure spaces Revista Mat. Iberoamericana 31 (2015), no. 2, 497-530.[35] J. M. Mazón, J. D. Rossi, and S. Segura de León, Functions of least gradient and 1-harmonic functions. Indiana Univ. Math. J. 63 (2014), no. 4, 1067-1084.[36] J. M. Mazón, J. D. Rossi, and S. Segura de León, The 1-Laplacian elliptic equation with inhomogeneous Robin boundary conditions, Differential and Integral Equations 28 (2015) no. 5-6, 401-430.[37] A. Mercaldo, J. D. Rossi, S. Segura de León, and C. Trombetti, Behaviour of p-Laplacian problems with Neumann boundary conditions when p goes to 1, Commun. Pure Appl. Anal. 12 (2013), 253-267.[38] M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J.Math. Pures Appl. (9) 82 (2003), no. 8, 975-1004.[39] A. Moradifam, Least gradient problemswith Neumann boundary condition, J. Differential Equations 263 (2017), no. 11, 7900-7918.ams inequality on metric measure spaces, Rev. Mat. Iberoam. 25 (2009), no. 2, 533-558.[40] T. Mäkäläinen, Adams inequality on metric measure spaces, Rev. Mat. Iberoam. 25 (2009), no. 2, 533-558.10.4171/RMI/575[41] C. Scheven and T. Schmidt, On the dual formulation of obstacle problems for the total variation and the area functional, to appear in Ann. Inst. Henri Poincaré, Anal. Non Linéaire. http://cvgmt.sns.it/paper/3320/[42] N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), no. 3, 1021-1050.[43] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16(2) (2000), 243-279.[44] P. Sternberg, G. Williams, and W. Ziemer, Existence, uniqueness, and regularity for functions of least gradient, J. Reine Angew. Math. 430 (1992), 35-60.[45] W. P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989.[46] W. Ziemer and K. Zumbrun, The obstacle problemfor functions of least gradient,Math. Bohem. 124 (1999), no. 2-3, 193-219.

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Analysis and Geometry in Metric Spacesde Gruyter

Published: Feb 16, 2018

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