# 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 and Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185
##### References:
 [1] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. [2] G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} =0$, Ark. Math., 7 (1968), 395-425. [3] G. Aronsson, On certain singular solutions of the partial differential equation $u_x^2 u_{xx} +2u_x u_y u_{xy}+ u_y^2 u_{yy} =0$, Manuscripta Mathematica, 47 (1984), 133-151. doi: 10.1007/BF01174590. [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 Proc. Japan -United States Sem., Tokyo (1977), 1-6. [5] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer-Verlag, 1991. doi: 10.1007/b97238. [6] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. of the American Mathematical Society (New series), 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3. [7] W. Blaschke, Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logaritmischen Potentials, Ber. Ver. Sächs. Akad. Wiss. Leipzig, 68 (1916), 3-7. [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 Some topics in nonlinear PDEs (Turin, 1989). Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue, 1568, 1991. [9] T. Carleman, Sur un problème d'unicité pour les systemes d'equations aux derivées partielles à deux variables indépendentes, Ark. for Mat., 26B (1939), 1-9. [10] M. G. Crandall, A visit with the $\infty$-Laplacian, in Calculus of variations and nonlinear partial differential equations, Lecture Notes in Mathematics, 1927 (2008), 75-122. doi: 10.1007/978-3-540-75914-0_3. [11] R. Courant and D. Hilbert, Methods of Mathematical Physics (Volume II), Interscience Publishers, 1962. [12] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13 (2001), 123-139. [13] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Processsing, 7 (1998), 376-386. doi: 10.1109/83.661188. [14] R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth Mathematics Series, 1984. [15] C. F. Gauss, Algemeine Lehrsätze in Beziehung auf die im verkehrtem Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abbstossungs-Kräfte, (1840), Werke, 5, Band, Göttingen, 1877. [16] N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015. [17] S. Granlund and N. Marola, On a frequency function approach to the unique continuation principle, Expo. Math., 30 (2012), 154-167. doi: 10.1016/j.exmath.2012.01.006. [18] S. Granlund and N. Marola, On the problem of unique continuation for the $p$-Laplace equation, Nonlinear Analysis, 101 (2014), 89-97. doi: 10.1016/j.na.2014.01.020. [19] F. Huckemann, On the "one circle" problem for harmonic functions, J. London Math. Soc., 29 (1954), 491-497. [20] W. Hansen and N. Nadirashvili, A converse to the mean value theorem for harmonic functions, Acta Math., 171 (1993), 139-163. doi: 10.1007/BF02392531. [21] W. Hansen and N. Nadirashvili, Littlewood's one circle problem, J. London Math. Soc., 50 (1994), 349-360. doi: 10.1016/j.exmath.2008.04.001. [22] R. Jensen, Uniqueness of lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368. [23] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schröinger operators, Ann. of Math., 12 (1985), 463-494. doi: 10.2307/1971205. [24] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179. [25] O. D. Kellogg, Converses of Gauss's theorem on the arithmetic mean, Trans. Amer. Math. Soc., 36 (1934), 227-242. doi: 10.2307/1989835. [26] B. Kawohl, J. J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, Journal des Mathéatiques Pures et Apliquées, 97 (2012) , 173-188. doi: 10.1016/j.matpur.2011.07.001. [27] C. Kenig, Carleman Estimates, uniform Sobolev Inequalities for second-order differential operators, and unique continuation theorems, in Proceedings of the International Congress of Mathematics, (Berkeley 1986). Vol. 1,2. Amer. Math. Soc. (1987), 948-960. [28] C. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, in Harmonic Analysis and Partial differential equations (El Escorial 1987). Lecture Notes in Math., 1384 (1989), 69-90. doi: 10.1007/BFb0086794. [29] P. Koebe, Herleitung der partiellen Differentialgleichungen der Potentialfunktion aus deren Integraleigenschaft, Sitzungsber. Berlin. Math. Gessellschaft, 5 (1906), 39-42. [30] J. E. Littlewood, Some Problems in Real and Complex Analysis, Hath. Math. Monographs, 1968. [31] F. H. Lin, A uniqueness theorem for parabolic equations, Comm. on Pure and Appl. Math., 43 (1990), 127-136. doi: 10.1002/cpa.3160430105. [32] P. Lindqvist, Notes on the $p$-Laplace equation, Report, University of Jyväkylä Department of Mathematics and Statistics, 102 (2006). [33] E. Le Gruyer, On absolutely minimizing lipschitz extension and PDE $\Delta_{\infty}(u) = 0$, Nonlinear Differential Equations and Applications, 14 (2007), 29-55. doi: 10.1007/s00030-006-4030-z. [34] J. G. Llorente, A note on unique continuation for solutions of the $\infty$-mean value property, Ann. Acad. Scient. Fennicae, 39 (2014), 473-483. doi: 10.5186/aasfm.2014.3914. [35] E. Le Gruyer and J. C. Archer, Harmonious extensions, Siam J. Math. Anal., 29 (1998), 279-292. doi: 10.1137/S0036141095294067. [36] H. Luiro, M. Parviainen and E. Saksman, On the existence and uniqueness of $p$-harmonious functions, Differential Integral Equations, 3-4 (2014), 201-216. [37] J. J. Manfredi, $p$-harmonic functions in the plane, Proc. Amer. Math. Soc., 103 (1988), 473-479. doi: 10.2307/2047164. [38] J. J. Manfredi, M. Parvianen and J. D. Rossi, An asymptotic mean value characterization for $p$-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1. [39] J. J. Manfredi, M. Parvianen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sc., 11 (2013), 215-241. [40] I. Netuka, J. Veselý, Mean value properties and harmonic functions, in Classical and modern potential theory and applications. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 430, Kluwer (1994), 359-398. [41] I. Privaloff, Sur les fonctions harmoniques, Rec. Math. Moscou (Mat. Sbornik), 32 (1925), 464-471. [42] Y. Peres, S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048. [43] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, Journal American Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1. [44] B. S. Thomson, Symmetric Properties of Real Functions, Marcel Dekker, 1994. [45] V. Volterra, Alcune osservazioni sopra propietá atte ad individuare una funzione, Rend. Acadd. d. Lincei Roma, 18 (1909), 263-266. [46] Y. Yu, A remark on $C^2$-infinity harmonic functions, Electronic J. of Differential Equations, 122 (2006), 1-4. [47] S. Zaremba, Contributions à la théorie d'une équation fonctionelle de la physique, Rend. Circ. Mat. Palermo, 19 (1905), 140-150.

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##### References:
 [1] G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561. [2] G. Aronsson, On the partial differential equation $u_x^2 u_{xx} + 2u_x u_y u_{xy} + u_y^2 u_{yy} =0$, Ark. Math., 7 (1968), 395-425. [3] G. Aronsson, On certain singular solutions of the partial differential equation $u_x^2 u_{xx} +2u_x u_y u_{xy}+ u_y^2 u_{yy} =0$, Manuscripta Mathematica, 47 (1984), 133-151. doi: 10.1007/BF01174590. [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 Proc. Japan -United States Sem., Tokyo (1977), 1-6. [5] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, Springer-Verlag, 1991. doi: 10.1007/b97238. [6] G. Aronsson, M. G. Crandall and P. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. of the American Mathematical Society (New series), 41 (2004), 439-505. doi: 10.1090/S0273-0979-04-01035-3. [7] W. Blaschke, Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logaritmischen Potentials, Ber. Ver. Sächs. Akad. Wiss. Leipzig, 68 (1916), 3-7. [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 Some topics in nonlinear PDEs (Turin, 1989). Rend. Sem. Mat. Univ. Politec. Torino 1989, Special Issue, 1568, 1991. [9] T. Carleman, Sur un problème d'unicité pour les systemes d'equations aux derivées partielles à deux variables indépendentes, Ark. for Mat., 26B (1939), 1-9. [10] M. G. Crandall, A visit with the $\infty$-Laplacian, in Calculus of variations and nonlinear partial differential equations, Lecture Notes in Mathematics, 1927 (2008), 75-122. doi: 10.1007/978-3-540-75914-0_3. [11] R. Courant and D. Hilbert, Methods of Mathematical Physics (Volume II), Interscience Publishers, 1962. [12] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations, 13 (2001), 123-139. [13] V. Caselles, J. M. Morel and C. Sbert, An axiomatic approach to image interpolation, IEEE Trans. Image Processsing, 7 (1998), 376-386. doi: 10.1109/83.661188. [14] R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth Mathematics Series, 1984. [15] C. F. Gauss, Algemeine Lehrsätze in Beziehung auf die im verkehrtem Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abbstossungs-Kräfte, (1840), Werke, 5, Band, Göttingen, 1877. [16] N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015. [17] S. Granlund and N. Marola, On a frequency function approach to the unique continuation principle, Expo. Math., 30 (2012), 154-167. doi: 10.1016/j.exmath.2012.01.006. [18] S. Granlund and N. Marola, On the problem of unique continuation for the $p$-Laplace equation, Nonlinear Analysis, 101 (2014), 89-97. doi: 10.1016/j.na.2014.01.020. [19] F. Huckemann, On the "one circle" problem for harmonic functions, J. London Math. Soc., 29 (1954), 491-497. [20] W. Hansen and N. Nadirashvili, A converse to the mean value theorem for harmonic functions, Acta Math., 171 (1993), 139-163. doi: 10.1007/BF02392531. [21] W. Hansen and N. Nadirashvili, Littlewood's one circle problem, J. London Math. Soc., 50 (1994), 349-360. doi: 10.1016/j.exmath.2008.04.001. [22] R. Jensen, Uniqueness of lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74. doi: 10.1007/BF00386368. [23] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schröinger operators, Ann. of Math., 12 (1985), 463-494. doi: 10.2307/1971205. [24] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear elliptic equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179. [25] O. D. Kellogg, Converses of Gauss's theorem on the arithmetic mean, Trans. Amer. Math. Soc., 36 (1934), 227-242. doi: 10.2307/1989835. [26] B. Kawohl, J. J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, Journal des Mathéatiques Pures et Apliquées, 97 (2012) , 173-188. doi: 10.1016/j.matpur.2011.07.001. [27] C. Kenig, Carleman Estimates, uniform Sobolev Inequalities for second-order differential operators, and unique continuation theorems, in Proceedings of the International Congress of Mathematics, (Berkeley 1986). Vol. 1,2. Amer. Math. Soc. (1987), 948-960. [28] C. Kenig, Restriction theorems, Carleman estimates, uniform Sobolev inequalities and unique continuation, in Harmonic Analysis and Partial differential equations (El Escorial 1987). Lecture Notes in Math., 1384 (1989), 69-90. doi: 10.1007/BFb0086794. [29] P. Koebe, Herleitung der partiellen Differentialgleichungen der Potentialfunktion aus deren Integraleigenschaft, Sitzungsber. Berlin. Math. Gessellschaft, 5 (1906), 39-42. [30] J. E. Littlewood, Some Problems in Real and Complex Analysis, Hath. Math. Monographs, 1968. [31] F. H. Lin, A uniqueness theorem for parabolic equations, Comm. on Pure and Appl. Math., 43 (1990), 127-136. doi: 10.1002/cpa.3160430105. [32] P. Lindqvist, Notes on the $p$-Laplace equation, Report, University of Jyväkylä Department of Mathematics and Statistics, 102 (2006). [33] E. Le Gruyer, On absolutely minimizing lipschitz extension and PDE $\Delta_{\infty}(u) = 0$, Nonlinear Differential Equations and Applications, 14 (2007), 29-55. doi: 10.1007/s00030-006-4030-z. [34] J. G. Llorente, A note on unique continuation for solutions of the $\infty$-mean value property, Ann. Acad. Scient. Fennicae, 39 (2014), 473-483. doi: 10.5186/aasfm.2014.3914. [35] E. Le Gruyer and J. C. Archer, Harmonious extensions, Siam J. Math. Anal., 29 (1998), 279-292. doi: 10.1137/S0036141095294067. [36] H. Luiro, M. Parviainen and E. Saksman, On the existence and uniqueness of $p$-harmonious functions, Differential Integral Equations, 3-4 (2014), 201-216. [37] J. J. Manfredi, $p$-harmonic functions in the plane, Proc. Amer. Math. Soc., 103 (1988), 473-479. doi: 10.2307/2047164. [38] J. J. Manfredi, M. Parvianen and J. D. Rossi, An asymptotic mean value characterization for $p$-harmonic functions, Proc. Amer. Math. Soc., 138 (2010), 881-889. doi: 10.1090/S0002-9939-09-10183-1. [39] J. J. Manfredi, M. Parvianen and J. D. Rossi, On the definition and properties of $p$-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sc., 11 (2013), 215-241. [40] I. Netuka, J. Veselý, Mean value properties and harmonic functions, in Classical and modern potential theory and applications. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 430, Kluwer (1994), 359-398. [41] I. Privaloff, Sur les fonctions harmoniques, Rec. Math. Moscou (Mat. Sbornik), 32 (1925), 464-471. [42] Y. Peres, S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048. [43] Y. Peres, O. Schramm, S. Sheffield and D. B. Wilson, Tug-of-war and the infinity Laplacian, Journal American Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1. [44] B. S. Thomson, Symmetric Properties of Real Functions, Marcel Dekker, 1994. [45] V. Volterra, Alcune osservazioni sopra propietá atte ad individuare una funzione, Rend. Acadd. d. Lincei Roma, 18 (1909), 263-266. [46] Y. Yu, A remark on $C^2$-infinity harmonic functions, Electronic J. of Differential Equations, 122 (2006), 1-4. [47] S. Zaremba, Contributions à la théorie d'une équation fonctionelle de la physique, Rend. Circ. Mat. Palermo, 19 (1905), 140-150.
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