# American Institute of Mathematical Sciences

January  2015, 14(1): 185-199. doi: 10.3934/cpaa.2015.14.185

## Mean value properties and unique continuation

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Received  January 2014 Revised  March 2014 Published  September 2014

In the first part of the paper we review some mean value properties and their connections to the Laplacian and other significant nonlinear operators like the $p$-Laplacian and the infinity-Laplacian. The second part is devoted to the unique continuation property, including a brief description of the methods, some of the main problems in the area and connections to the so called infinity mean value property.
Citation: José G. Llorente. Mean value properties and unique continuation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185
##### References:
 [1] G. Aronsson, Extension of functions satisfying Lipschitz conditions,, \emph{Ark. Mat.}, 6 (1967), 551. Google Scholar [2] G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} =0$,, \emph{Ark. Math.}, 7 (1968), 395. Google Scholar [3] G. Aronsson, On certain singular solutions of the partial differential equation $u_x^2 u_{xx} +2u_x u_y u_{xy}+ u_y^2 u_{yy} =0$,, \emph{Manuscripta Mathematica}, 47 (1984), 133. doi: 10.1007/BF01174590. Google Scholar [4] F. J. Jr. Almgrem, Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, minimal submanifolds and geodesics,, in \emph{Proc. Japan -United States Sem.}, (1977), 1. Google Scholar [5] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory,, Springer-Verlag, (1991). doi: 10.1007/b97238. Google Scholar [6] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions,, \emph{Bull. of the American Mathematical Society (New series)}, 41 (2004), 439. doi: 10.1090/S0273-0979-04-01035-3. Google Scholar [7] W. Blaschke, Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logaritmischen Potentials,, \emph{Ber. Ver. S\, 68 (1916), 3. Google Scholar [8] T. Bhattacharya, E. DiBenedetto and J. J. Manfredi, Limits as $p\to \infty$ of $\Delta_p u_p = f$ and and related extremal problems,, in \emph{Some topics in nonlinear PDEs (Turin, 1568 (1989). Google Scholar [9] T. Carleman, Sur un problème d'unicité pour les systemes d'equations aux derivées partielles à deux variables indépendentes,, \emph{Ark. for Mat.}, 26B (1939), 1. Google Scholar [10] M. G. Crandall, A visit with the $\infty$-Laplacian,, in \emph{Calculus of variations and nonlinear partial differential equations}, 1927 (2008), 75. doi: 10.1007/978-3-540-75914-0_3. Google Scholar [11] R. Courant and D. Hilbert, Methods of Mathematical Physics (Volume II),, Interscience Publishers, (1962). Google Scholar [12] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 13 (2001), 123. Google Scholar [13] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation,, \emph{IEEE Trans. Image Processsing}, 7 (1998), 376. doi: 10.1109/83.661188. Google Scholar [14] R. Durrett, Brownian Motion and Martingales in Analysis,, Wadsworth Mathematics Series, (1984). Google Scholar [15] C. F. Gauss, Algemeine Lehrsätze in Beziehung auf die im verkehrtem Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abbstossungs-Kräfte,, (1840), (1840). Google Scholar [16] N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation,, \emph{Indiana Univ. Math. J.}, 35 (1986), 245. doi: 10.1512/iumj.1986.35.35015. Google Scholar [17] S. Granlund and N. Marola, On a frequency function approach to the unique continuation principle,, \emph{Expo. Math.}, 30 (2012), 154. doi: 10.1016/j.exmath.2012.01.006. Google Scholar [18] S. Granlund and N. Marola, On the problem of unique continuation for the $p$-Laplace equation,, \emph{Nonlinear Analysis}, 101 (2014), 89. doi: 10.1016/j.na.2014.01.020. Google Scholar [19] F. Huckemann, On the "one circle" problem for harmonic functions,, \emph{J. London Math. Soc.}, 29 (1954), 491. Google Scholar [20] W. Hansen and N. Nadirashvili, A converse to the mean value theorem for harmonic functions,, \emph{Acta Math.}, 171 (1993), 139. doi: 10.1007/BF02392531. Google Scholar [21] W. Hansen and N. Nadirashvili, Littlewood's one circle problem,, \emph{J. London Math. Soc.}, 50 (1994), 349. doi: 10.1016/j.exmath.2008.04.001. Google Scholar [22] R. Jensen, Uniqueness of lipschitz extensions: minimizing the sup norm of the gradient,, \emph{Arch. Rational Mech. Anal.}, 123 (1993), 51. doi: 10.1007/BF00386368. Google Scholar [23] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schröinger operators,, \emph{Ann. of Math.}, 12 (1985), 463. doi: 10.2307/1971205. Google Scholar [24] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699. doi: 10.1137/S0036141000372179. Google Scholar [25] O. D. Kellogg, Converses of Gauss's theorem on the arithmetic mean,, \emph{Trans. Amer. Math. Soc.}, 36 (1934), 227. doi: 10.2307/1989835. Google Scholar [26] B. Kawohl, J. J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages,, \emph{Journal des Math\'eatiques Pures et Apliqu\'ees}, 97 (2012), 173. doi: 10.1016/j.matpur.2011.07.001. Google Scholar [27] C. Kenig, Carleman Estimates, uniform Sobolev Inequalities for second-order differential operators, and unique continuation theorems,, in \emph{Proceedings of the International Congress of Mathematics}, (1987), 948. Google Scholar [28] C. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation,, in \emph{Harmonic Analysis and Partial differential equations} (El Escorial 1987). Lecture Notes in Math., 1384 (1989), 69. doi: 10.1007/BFb0086794. Google Scholar [29] P. Koebe, Herleitung der partiellen Differentialgleichungen der Potentialfunktion aus deren Integraleigenschaft,, Sitzungsber. Berlin. Math. Gessellschaft, 5 (1906), 39. Google Scholar [30] J. E. Littlewood, Some Problems in Real and Complex Analysis,, Hath. Math. Monographs, (1968). Google Scholar [31] F. H. Lin, A uniqueness theorem for parabolic equations,, \emph{Comm. on Pure and Appl. Math.}, 43 (1990), 127. doi: 10.1002/cpa.3160430105. Google Scholar [32] P. Lindqvist, Notes on the $p$-Laplace equation,, Report, 102 (2006). Google Scholar [33] E. Le Gruyer, On absolutely minimizing lipschitz extension and PDE $\Delta_{\infty}(u) = 0$,, \emph{Nonlinear Differential Equations and Applications}, 14 (2007), 29. doi: 10.1007/s00030-006-4030-z. Google Scholar [34] J. G. Llorente, A note on unique continuation for solutions of the $\infty$-mean value property,, \emph{Ann. Acad. Scient. Fennicae}, 39 (2014), 473. doi: 10.5186/aasfm.2014.3914. Google Scholar [35] E. Le Gruyer and J. C. Archer, Harmonious extensions,, \emph{Siam J. Math. Anal.}, 29 (1998), 279. doi: 10.1137/S0036141095294067. Google Scholar [36] H. Luiro, M. Parviainen and E. Saksman, On the existence and uniqueness of $p$-harmonious functions,, \emph{Differential Integral Equations}, 3-4 (2014), 3. Google Scholar [37] J. J. Manfredi, $p$-harmonic functions in the plane,, \emph{Proc. Amer. Math. Soc.}, 103 (1988), 473. doi: 10.2307/2047164. Google Scholar [38] J. J. Manfredi, M. Parvianen and J. D. Rossi, An asymptotic mean value characterization for $p$-harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 881. doi: 10.1090/S0002-9939-09-10183-1. Google Scholar [39] J. J. Manfredi, M. Parvianen and J. D. Rossi, On the definition and properties of $p$-harmonious functions,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sc.}, 11 (2013), 215. Google Scholar [40] I. Netuka, J. Veselý, Mean value properties and harmonic functions,, in \emph{Classical and modern potential theory and applications. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.}, 430 (1994), 359. Google Scholar [41] I. Privaloff, Sur les fonctions harmoniques,, \emph{Rec. Math. Moscou (Mat. Sbornik)}, 32 (1925), 464. Google Scholar [42] Y. Peres, S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian,, \emph{Duke Math. J.}, 145 (2008), 91. doi: 10.1215/00127094-2008-048. Google Scholar [43] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian,, \emph{Journal American Math. Soc.}, 22 (2009), 167. doi: 10.1090/S0894-0347-08-00606-1. Google Scholar [44] B. S. Thomson, Symmetric Properties of Real Functions,, Marcel Dekker, (1994). Google Scholar [45] V. Volterra, Alcune osservazioni sopra propietá atte ad individuare una funzione,, \emph{Rend. Acadd. d. Lincei Roma}, 18 (1909), 263. Google Scholar [46] Y. Yu, A remark on $C^2$-infinity harmonic functions,, \emph{Electronic J. of Differential Equations}, 122 (2006), 1. Google Scholar [47] S. Zaremba, Contributions à la théorie d'une équation fonctionelle de la physique,, \emph{Rend. Circ. Mat. Palermo}, 19 (1905), 140. Google Scholar

show all references

##### References:
 [1] G. Aronsson, Extension of functions satisfying Lipschitz conditions,, \emph{Ark. Mat.}, 6 (1967), 551. Google Scholar [2] G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} =0$,, \emph{Ark. Math.}, 7 (1968), 395. Google Scholar [3] G. Aronsson, On certain singular solutions of the partial differential equation $u_x^2 u_{xx} +2u_x u_y u_{xy}+ u_y^2 u_{yy} =0$,, \emph{Manuscripta Mathematica}, 47 (1984), 133. doi: 10.1007/BF01174590. Google Scholar [4] F. J. Jr. Almgrem, Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, minimal submanifolds and geodesics,, in \emph{Proc. Japan -United States Sem.}, (1977), 1. Google Scholar [5] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory,, Springer-Verlag, (1991). doi: 10.1007/b97238. Google Scholar [6] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions,, \emph{Bull. of the American Mathematical Society (New series)}, 41 (2004), 439. doi: 10.1090/S0273-0979-04-01035-3. Google Scholar [7] W. Blaschke, Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logaritmischen Potentials,, \emph{Ber. Ver. S\, 68 (1916), 3. Google Scholar [8] T. Bhattacharya, E. DiBenedetto and J. J. Manfredi, Limits as $p\to \infty$ of $\Delta_p u_p = f$ and and related extremal problems,, in \emph{Some topics in nonlinear PDEs (Turin, 1568 (1989). Google Scholar [9] T. Carleman, Sur un problème d'unicité pour les systemes d'equations aux derivées partielles à deux variables indépendentes,, \emph{Ark. for Mat.}, 26B (1939), 1. Google Scholar [10] M. G. Crandall, A visit with the $\infty$-Laplacian,, in \emph{Calculus of variations and nonlinear partial differential equations}, 1927 (2008), 75. doi: 10.1007/978-3-540-75914-0_3. Google Scholar [11] R. Courant and D. Hilbert, Methods of Mathematical Physics (Volume II),, Interscience Publishers, (1962). Google Scholar [12] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 13 (2001), 123. Google Scholar [13] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation,, \emph{IEEE Trans. Image Processsing}, 7 (1998), 376. doi: 10.1109/83.661188. Google Scholar [14] R. Durrett, Brownian Motion and Martingales in Analysis,, Wadsworth Mathematics Series, (1984). Google Scholar [15] C. F. Gauss, Algemeine Lehrsätze in Beziehung auf die im verkehrtem Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abbstossungs-Kräfte,, (1840), (1840). Google Scholar [16] N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation,, \emph{Indiana Univ. Math. J.}, 35 (1986), 245. doi: 10.1512/iumj.1986.35.35015. Google Scholar [17] S. Granlund and N. Marola, On a frequency function approach to the unique continuation principle,, \emph{Expo. Math.}, 30 (2012), 154. doi: 10.1016/j.exmath.2012.01.006. Google Scholar [18] S. Granlund and N. Marola, On the problem of unique continuation for the $p$-Laplace equation,, \emph{Nonlinear Analysis}, 101 (2014), 89. doi: 10.1016/j.na.2014.01.020. Google Scholar [19] F. Huckemann, On the "one circle" problem for harmonic functions,, \emph{J. London Math. Soc.}, 29 (1954), 491. Google Scholar [20] W. Hansen and N. Nadirashvili, A converse to the mean value theorem for harmonic functions,, \emph{Acta Math.}, 171 (1993), 139. doi: 10.1007/BF02392531. Google Scholar [21] W. Hansen and N. Nadirashvili, Littlewood's one circle problem,, \emph{J. London Math. Soc.}, 50 (1994), 349. doi: 10.1016/j.exmath.2008.04.001. Google Scholar [22] R. Jensen, Uniqueness of lipschitz extensions: minimizing the sup norm of the gradient,, \emph{Arch. Rational Mech. Anal.}, 123 (1993), 51. doi: 10.1007/BF00386368. Google Scholar [23] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schröinger operators,, \emph{Ann. of Math.}, 12 (1985), 463. doi: 10.2307/1971205. Google Scholar [24] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation,, \emph{SIAM J. Math. Anal.}, 33 (2001), 699. doi: 10.1137/S0036141000372179. Google Scholar [25] O. D. Kellogg, Converses of Gauss's theorem on the arithmetic mean,, \emph{Trans. Amer. Math. Soc.}, 36 (1934), 227. doi: 10.2307/1989835. Google Scholar [26] B. Kawohl, J. J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages,, \emph{Journal des Math\'eatiques Pures et Apliqu\'ees}, 97 (2012), 173. doi: 10.1016/j.matpur.2011.07.001. Google Scholar [27] C. Kenig, Carleman Estimates, uniform Sobolev Inequalities for second-order differential operators, and unique continuation theorems,, in \emph{Proceedings of the International Congress of Mathematics}, (1987), 948. Google Scholar [28] C. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation,, in \emph{Harmonic Analysis and Partial differential equations} (El Escorial 1987). Lecture Notes in Math., 1384 (1989), 69. doi: 10.1007/BFb0086794. Google Scholar [29] P. Koebe, Herleitung der partiellen Differentialgleichungen der Potentialfunktion aus deren Integraleigenschaft,, Sitzungsber. Berlin. Math. Gessellschaft, 5 (1906), 39. Google Scholar [30] J. E. Littlewood, Some Problems in Real and Complex Analysis,, Hath. Math. Monographs, (1968). Google Scholar [31] F. H. Lin, A uniqueness theorem for parabolic equations,, \emph{Comm. on Pure and Appl. Math.}, 43 (1990), 127. doi: 10.1002/cpa.3160430105. Google Scholar [32] P. Lindqvist, Notes on the $p$-Laplace equation,, Report, 102 (2006). Google Scholar [33] E. Le Gruyer, On absolutely minimizing lipschitz extension and PDE $\Delta_{\infty}(u) = 0$,, \emph{Nonlinear Differential Equations and Applications}, 14 (2007), 29. doi: 10.1007/s00030-006-4030-z. Google Scholar [34] J. G. Llorente, A note on unique continuation for solutions of the $\infty$-mean value property,, \emph{Ann. Acad. Scient. Fennicae}, 39 (2014), 473. doi: 10.5186/aasfm.2014.3914. Google Scholar [35] E. Le Gruyer and J. C. Archer, Harmonious extensions,, \emph{Siam J. Math. Anal.}, 29 (1998), 279. doi: 10.1137/S0036141095294067. Google Scholar [36] H. Luiro, M. Parviainen and E. Saksman, On the existence and uniqueness of $p$-harmonious functions,, \emph{Differential Integral Equations}, 3-4 (2014), 3. Google Scholar [37] J. J. Manfredi, $p$-harmonic functions in the plane,, \emph{Proc. Amer. Math. Soc.}, 103 (1988), 473. doi: 10.2307/2047164. Google Scholar [38] J. J. Manfredi, M. Parvianen and J. D. Rossi, An asymptotic mean value characterization for $p$-harmonic functions,, \emph{Proc. Amer. Math. Soc.}, 138 (2010), 881. doi: 10.1090/S0002-9939-09-10183-1. Google Scholar [39] J. J. Manfredi, M. Parvianen and J. D. Rossi, On the definition and properties of $p$-harmonious functions,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sc.}, 11 (2013), 215. Google Scholar [40] I. Netuka, J. Veselý, Mean value properties and harmonic functions,, in \emph{Classical and modern potential theory and applications. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.}, 430 (1994), 359. Google Scholar [41] I. Privaloff, Sur les fonctions harmoniques,, \emph{Rec. Math. Moscou (Mat. Sbornik)}, 32 (1925), 464. Google Scholar [42] Y. Peres, S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian,, \emph{Duke Math. J.}, 145 (2008), 91. doi: 10.1215/00127094-2008-048. Google Scholar [43] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian,, \emph{Journal American Math. Soc.}, 22 (2009), 167. doi: 10.1090/S0894-0347-08-00606-1. Google Scholar [44] B. S. Thomson, Symmetric Properties of Real Functions,, Marcel Dekker, (1994). Google Scholar [45] V. Volterra, Alcune osservazioni sopra propietá atte ad individuare una funzione,, \emph{Rend. Acadd. d. Lincei Roma}, 18 (1909), 263. Google Scholar [46] Y. Yu, A remark on $C^2$-infinity harmonic functions,, \emph{Electronic J. of Differential Equations}, 122 (2006), 1. Google Scholar [47] S. Zaremba, Contributions à la théorie d'une équation fonctionelle de la physique,, \emph{Rend. Circ. Mat. Palermo}, 19 (1905), 140. Google Scholar
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