Mean value properties and unique continuation

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

Mean value properties and unique continuation

José G. Llorente ^1,

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

Received January 2014 Revised March 2014 Published September 2014

Abstract
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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.

Keywords: Mean value property, unique continuation., infinity Laplacian.

Mathematics Subject Classification: Primary: 31C05, 35B60; Secondary: 31C4.

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.

show all references

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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