August  2019, 39(8): 4487-4518. doi: 10.3934/dcds.2019184

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

1. 

Dipartimento di Scienze di Base e Applicate per I'Ingegneria, Sapienza Università di Roma Roma, Italy, Via A. Scarpa 16, 00161 Roma, Italy

2. 

Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Mayagüez, Puerto Rico, 00681, USA

* Corresponding author: A. Vélez-Santiago

Received  August 2018 Published  May 2019

We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

Citation: Maria Rosaria Lancia, Alejandro Vélez-Santiago, Paola Vernole. A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4487-4518. doi: 10.3934/dcds.2019184
References:
[1]

R. Adams, Sobolev Spaces, New York-London, 1975.

[2]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Sublinear and superlinear Ambrosetti–Prodi problems for the Dirichlet $p$-Laplacian, Nonlinear Analysis, 95 (2014), 263-280. doi: 10.1016/j.na.2013.08.026.

[3]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Annali Mat. Pura Appl., 93 (1972), 231-246. doi: 10.1007/BF02412022.

[4]

D. Arcoya and D. Ruiz, The Ambrosetti–Prodi problem for the p-Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447.

[5]

M. Arias and M. Cuesta, A one side superlinear Ambrosetti–Prodi problem for the Dirichlet $p$-laplacian, J. Math. Anal. Appl., 367 (2010), 499-507. doi: 10.1016/j.jmaa.2010.01.031.

[6]

M. Biegert, A priori estimate for the difference of solutions to quasi-linear elliptic equations, Manuscripta Math., 133 (2010), 273-306. doi: 10.1007/s00229-010-0367-z.

[7]

M. Biegert, On trace of Sobolev functions on the boundary of extension domains, Proc. Amer. Math. Soc., 137 (2009), 4169-4176. doi: 10.1090/S0002-9939-09-10045-X.

[8]

M. Biegert and P. Vernole, Strongly local nonlinear Dirichlet functionals and forms, Adv. Math. Sci. Appl., 15 (2005), 655-682.

[9]

V. I. Burenkov, Sobolev Spaces on Domains, TEUBNER-TEXTE zur Mathematik, Vol. 137, 1998. doi: 10.1007/978-3-663-11374-4.

[10]

R. Capitanelli, Homogeneous p-Lagrangians and self-similarity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 215-235.

[11]

R. Capitanelli, Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Anal., 3 (2002), 67-80.

[12]

R. Capitanelli and M. R. Lancia, Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. Convex Anal., 9 (2002), 245-257.

[13]

M. CefaloG. Dell'acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, AMC, 218 (2012), 5453-5473. doi: 10.1016/j.amc.2011.11.033.

[14]

M. Cefalo and M. R. Lancia, An optimal mesh generation algorithm for domains with Koch type boundaries, Math. Comput. Simulation, 106 (2014), 133-162. doi: 10.1016/j.matcom.2014.04.009.

[15]

M. CefaloM. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differential and Integral equations, 26 (2013), 1027-1054.

[16]

S. CreoM. R. LanciaA. Vélez-Santiago and P. Vernole, Approximation of a nonlinear fractal energy functional on varying Hilbert spaces, Comm. Pure Appl. Anal., 17 (2018), 647-669. doi: 10.3934/cpaa.2018035.

[17]

S. Creo and V. Regis Durante, Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 65-90. doi: 10.3934/dcdss.2019005.

[18]

D. Danielli, N. Garofalo and D.-H. Nhieu, Non-doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces, Mem. Amer. Math. Soc., 182 (2006), x+119 pp. doi: 10.1090/memo/0857.

[19]

F. O.de Paiva and M. Montenegro, An Ambrosetti-Prodi-type result for a quasilinear Neumann problem, Proc. Edinburgh Math. Soc., 55 (2012), 771-780. doi: 10.1017/S0013091512000041.

[20]

F. O. de Paiva and A. E. Presoto, A Neumann problem of Ambrosetti-Prodi type, J. Fixed Point Theory Appl., 18 (2016), 189-200. doi: 10.1007/s11784-015-0277-5.

[21]

U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen., 23 (2004), 115-137. doi: 10.4171/ZAA/1190.

[22]

M. Fukushima, Y. Oshima and M. Takeda,, Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, vol. 19, Berlin Eds. Bauer Kazdan, Zehnder, 1994. doi: 10.1515/9783110889741.

[23]

P. HajłaszP. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Functional Analysis, 254 (2008), 1217-1234. doi: 10.1016/j.jfa.2007.11.020.

[24]

A. Jonsson and H. Wallin, Function Spaces on Subsets of  $\mathbb{R\!}^{\, n} $., Math. Rep. Vol. 2 Part Ⅰ, Academic Publisher, Harwood, 1984.

[25]

E. Koizumi and K. Schmitt, Ambrosetti–Prodi-type problems for quasilinear elliptic problems, Diff. Integ. Equations, 18 (2005), 241-262.

[26]

O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, in: Mathematics in Science and Engineering, 46. Academic Press, New York-London, 1968.

[27]

M. R. LanciaA. Vélez-Santiago and P. Vernole, Quasi-linear Venttsel' problems with nonlocal boundary conditions on fractal domains, Nonlinear Anal. Real Worl Appl., 35 (2017), 265-291. doi: 10.1016/j.nonrwa.2016.11.002.

[28]

M. R. Lancia and P. Vernole, Semilinear Venttsel problems in fractal domains, Applied Mathematics, 5 (2014), 1820-1833.

[29]

M. R. Lancia and P. Vernole, Venttsel problems in fractal domains, J. Evolution Equations, 14 (2014), 681-712. doi: 10.1007/s00028-014-0233-7.

[30]

M. R. LanciaV. Regis Durante and P. Vernole, Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1493-1520. doi: 10.3934/dcdss.2016060.

[31]

M. R. Lancia and M. A. Vivaldi, Lipschitz spaces and Besov traces on self-similar fractals, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 23 (1999), 101-116.

[32]

V. Lappalainen and A. Lehtonen, Embedding of Orliz-Sobolev spaces in Hölder spaces, Annales Academiæ Scientiarum Fennicæ, 14 (1989), 41-46. doi: 10.5186/aasfm.1989.1417.

[33]

V. K. Le, On a sub-supersolutions method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Analysis, 71 (2009), 3305-3321. doi: 10.1016/j.na.2009.01.211.

[34]

V. K. Le and K. Schmitt, Some general concepts of sub- and supersolutions for nonlinear elliptic problems, Topol. Methods Nonlinear Anal., 28 (2006), 87-103.

[35]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[36]

J. Maly and U. Mosco, Remarks on measure-valued Lagrangians on homogeneous spaces, Ricerche Mat., 48 (1999), 217-231.

[37]

J. Mawhin, Ambrosetti–Prodi-type results in nonlinear boundary value problems, Lecture Notes in Mathematics, 1285 (1987), 290-313. doi: 10.1007/BFb0080609.

[38]

V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3.

[39]

T. J. Miotto, Superlinear Ambrosetti–Prodi problem for the p-Laplacian operator, Nonlinear Differ. Equ. Appl. NoDEA, 17 (2010), 337-353. doi: 10.1007/s00030-010-0057-2.

[40]

T. J. Miotto, On the Ambrosetti-Prodi problem for a system involving p-Laplacian operator, Math Nachr., 289 (2016), 67-84. doi: 10.1002/mana.201300350.

[41]

U. Mosco, Lagrangian metrics on fractals, Proc. Symp. Appl. Math., 54, Amer. Math. Soc., R.Spigler and S. Venakides eds., (1998), 301–323.

[42]

M. R. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Annali Mat. Pura Appl., 80 (1968), 1-122. doi: 10.1007/BF02413623.

[43]

J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.

[44]

D. D. Repovš, Ambrosetti-Prodi problem with degenerate potential and Neumann boundary conditions, Electronic J. Differential Equations, 2018 (2018), Paper No. 41, 10 pp.

[45]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997.

[46]

E. Sovrano, Ambrosetti-Prodi type result to a Neumann problem via a topological approach, Discrete Contin. Dyn. Syst. Ser. S, 11 (2009), 345-355. doi: 10.3934/dcdss.2018019.

[47]

A. Vélez-Santiago, A quasi-linear Neumann problem of Ambrosetti–Prodi type on extension domains, Nonlinear Analysis, 160 (2017), 191-210. doi: 10.1016/j.na.2017.05.012.

[48]

A. Vélez-Santiago, Ambrosetti–Prodi-type problems for quasi-linear elliptic equations with nonlocal boundary conditions, Calc. Var. PDEs, 54 (2015), 3439-3469. doi: 10.1007/s00526-015-0910-6.

[49]

A. Vélez-Santiago, Global regularity for a class of quasi-linear local and nonlocal elliptic equations on extension domains, J. Functional Analysis, 269 (2015), 1-46. doi: 10.1016/j.jfa.2015.04.016.

[50]

A. Vélez-Santiago and M. Warma, A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions, J. Math. Anal. Appl., 372 (2010), 120-139. doi: 10.1016/j.jmaa.2010.07.003.

[51]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125. doi: 10.1007/BF02567633.

[52]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neuman and Robin boundary conditions on open sets, Potential Analysis, 42 (2015), 499-547. doi: 10.1007/s11118-014-9443-4.

[53]

W. P. Ziemer, Weakly Differentiable Functions, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:
[1]

R. Adams, Sobolev Spaces, New York-London, 1975.

[2]

S. AizicoviciN. S. Papageorgiou and V. Staicu, Sublinear and superlinear Ambrosetti–Prodi problems for the Dirichlet $p$-Laplacian, Nonlinear Analysis, 95 (2014), 263-280. doi: 10.1016/j.na.2013.08.026.

[3]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Annali Mat. Pura Appl., 93 (1972), 231-246. doi: 10.1007/BF02412022.

[4]

D. Arcoya and D. Ruiz, The Ambrosetti–Prodi problem for the p-Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865. doi: 10.1080/03605300500394447.

[5]

M. Arias and M. Cuesta, A one side superlinear Ambrosetti–Prodi problem for the Dirichlet $p$-laplacian, J. Math. Anal. Appl., 367 (2010), 499-507. doi: 10.1016/j.jmaa.2010.01.031.

[6]

M. Biegert, A priori estimate for the difference of solutions to quasi-linear elliptic equations, Manuscripta Math., 133 (2010), 273-306. doi: 10.1007/s00229-010-0367-z.

[7]

M. Biegert, On trace of Sobolev functions on the boundary of extension domains, Proc. Amer. Math. Soc., 137 (2009), 4169-4176. doi: 10.1090/S0002-9939-09-10045-X.

[8]

M. Biegert and P. Vernole, Strongly local nonlinear Dirichlet functionals and forms, Adv. Math. Sci. Appl., 15 (2005), 655-682.

[9]

V. I. Burenkov, Sobolev Spaces on Domains, TEUBNER-TEXTE zur Mathematik, Vol. 137, 1998. doi: 10.1007/978-3-663-11374-4.

[10]

R. Capitanelli, Homogeneous p-Lagrangians and self-similarity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 27 (2003), 215-235.

[11]

R. Capitanelli, Nonlinear energy forms on certain fractal curves, J. Nonlinear Convex Anal., 3 (2002), 67-80.

[12]

R. Capitanelli and M. R. Lancia, Nonlinear energy forms and Lipschitz spaces on the Koch curve, J. Convex Anal., 9 (2002), 245-257.

[13]

M. CefaloG. Dell'acqua and M. R. Lancia, Numerical approximation of transmission problems across Koch-type highly conductive layers, AMC, 218 (2012), 5453-5473. doi: 10.1016/j.amc.2011.11.033.

[14]

M. Cefalo and M. R. Lancia, An optimal mesh generation algorithm for domains with Koch type boundaries, Math. Comput. Simulation, 106 (2014), 133-162. doi: 10.1016/j.matcom.2014.04.009.

[15]

M. CefaloM. R. Lancia and H. Liang, Heat flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differential and Integral equations, 26 (2013), 1027-1054.

[16]

S. CreoM. R. LanciaA. Vélez-Santiago and P. Vernole, Approximation of a nonlinear fractal energy functional on varying Hilbert spaces, Comm. Pure Appl. Anal., 17 (2018), 647-669. doi: 10.3934/cpaa.2018035.

[17]

S. Creo and V. Regis Durante, Convergence and density results for parabolic quasi-linear Venttsel' problems in fractal domains, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 65-90. doi: 10.3934/dcdss.2019005.

[18]

D. Danielli, N. Garofalo and D.-H. Nhieu, Non-doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces, Mem. Amer. Math. Soc., 182 (2006), x+119 pp. doi: 10.1090/memo/0857.

[19]

F. O.de Paiva and M. Montenegro, An Ambrosetti-Prodi-type result for a quasilinear Neumann problem, Proc. Edinburgh Math. Soc., 55 (2012), 771-780. doi: 10.1017/S0013091512000041.

[20]

F. O. de Paiva and A. E. Presoto, A Neumann problem of Ambrosetti-Prodi type, J. Fixed Point Theory Appl., 18 (2016), 189-200. doi: 10.1007/s11784-015-0277-5.

[21]

U. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen., 23 (2004), 115-137. doi: 10.4171/ZAA/1190.

[22]

M. Fukushima, Y. Oshima and M. Takeda,, Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, vol. 19, Berlin Eds. Bauer Kazdan, Zehnder, 1994. doi: 10.1515/9783110889741.

[23]

P. HajłaszP. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Functional Analysis, 254 (2008), 1217-1234. doi: 10.1016/j.jfa.2007.11.020.

[24]

A. Jonsson and H. Wallin, Function Spaces on Subsets of  $\mathbb{R\!}^{\, n} $., Math. Rep. Vol. 2 Part Ⅰ, Academic Publisher, Harwood, 1984.

[25]

E. Koizumi and K. Schmitt, Ambrosetti–Prodi-type problems for quasilinear elliptic problems, Diff. Integ. Equations, 18 (2005), 241-262.

[26]

O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, in: Mathematics in Science and Engineering, 46. Academic Press, New York-London, 1968.

[27]

M. R. LanciaA. Vélez-Santiago and P. Vernole, Quasi-linear Venttsel' problems with nonlocal boundary conditions on fractal domains, Nonlinear Anal. Real Worl Appl., 35 (2017), 265-291. doi: 10.1016/j.nonrwa.2016.11.002.

[28]

M. R. Lancia and P. Vernole, Semilinear Venttsel problems in fractal domains, Applied Mathematics, 5 (2014), 1820-1833.

[29]

M. R. Lancia and P. Vernole, Venttsel problems in fractal domains, J. Evolution Equations, 14 (2014), 681-712. doi: 10.1007/s00028-014-0233-7.

[30]

M. R. LanciaV. Regis Durante and P. Vernole, Asymptotics for Venttsel' problems for operators in non divergence form in irregular domains, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1493-1520. doi: 10.3934/dcdss.2016060.

[31]

M. R. Lancia and M. A. Vivaldi, Lipschitz spaces and Besov traces on self-similar fractals, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 23 (1999), 101-116.

[32]

V. Lappalainen and A. Lehtonen, Embedding of Orliz-Sobolev spaces in Hölder spaces, Annales Academiæ Scientiarum Fennicæ, 14 (1989), 41-46. doi: 10.5186/aasfm.1989.1417.

[33]

V. K. Le, On a sub-supersolutions method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Analysis, 71 (2009), 3305-3321. doi: 10.1016/j.na.2009.01.211.

[34]

V. K. Le and K. Schmitt, Some general concepts of sub- and supersolutions for nonlinear elliptic problems, Topol. Methods Nonlinear Anal., 28 (2006), 87-103.

[35]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

[36]

J. Maly and U. Mosco, Remarks on measure-valued Lagrangians on homogeneous spaces, Ricerche Mat., 48 (1999), 217-231.

[37]

J. Mawhin, Ambrosetti–Prodi-type results in nonlinear boundary value problems, Lecture Notes in Mathematics, 1285 (1987), 290-313. doi: 10.1007/BFb0080609.

[38]

V. G. Maz'ya, Sobolev Spaces, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-09922-3.

[39]

T. J. Miotto, Superlinear Ambrosetti–Prodi problem for the p-Laplacian operator, Nonlinear Differ. Equ. Appl. NoDEA, 17 (2010), 337-353. doi: 10.1007/s00030-010-0057-2.

[40]

T. J. Miotto, On the Ambrosetti-Prodi problem for a system involving p-Laplacian operator, Math Nachr., 289 (2016), 67-84. doi: 10.1002/mana.201300350.

[41]

U. Mosco, Lagrangian metrics on fractals, Proc. Symp. Appl. Math., 54, Amer. Math. Soc., R.Spigler and S. Venakides eds., (1998), 301–323.

[42]

M. R. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Annali Mat. Pura Appl., 80 (1968), 1-122. doi: 10.1007/BF02413623.

[43]

J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.

[44]

D. D. Repovš, Ambrosetti-Prodi problem with degenerate potential and Neumann boundary conditions, Electronic J. Differential Equations, 2018 (2018), Paper No. 41, 10 pp.

[45]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1997.

[46]

E. Sovrano, Ambrosetti-Prodi type result to a Neumann problem via a topological approach, Discrete Contin. Dyn. Syst. Ser. S, 11 (2009), 345-355. doi: 10.3934/dcdss.2018019.

[47]

A. Vélez-Santiago, A quasi-linear Neumann problem of Ambrosetti–Prodi type on extension domains, Nonlinear Analysis, 160 (2017), 191-210. doi: 10.1016/j.na.2017.05.012.

[48]

A. Vélez-Santiago, Ambrosetti–Prodi-type problems for quasi-linear elliptic equations with nonlocal boundary conditions, Calc. Var. PDEs, 54 (2015), 3439-3469. doi: 10.1007/s00526-015-0910-6.

[49]

A. Vélez-Santiago, Global regularity for a class of quasi-linear local and nonlocal elliptic equations on extension domains, J. Functional Analysis, 269 (2015), 1-46. doi: 10.1016/j.jfa.2015.04.016.

[50]

A. Vélez-Santiago and M. Warma, A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions, J. Math. Anal. Appl., 372 (2010), 120-139. doi: 10.1016/j.jmaa.2010.07.003.

[51]

H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math., 73 (1991), 117-125. doi: 10.1007/BF02567633.

[52]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neuman and Robin boundary conditions on open sets, Potential Analysis, 42 (2015), 499-547. doi: 10.1007/s11118-014-9443-4.

[53]

W. P. Ziemer, Weakly Differentiable Functions, Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

Figure 1.  Pre-fractal Koch snowflake
Figure 2.  Surface S3
[1]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101

[2]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731

[3]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258

[4]

Michel L. Lapidus, Robert G. Niemeyer. Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3719-3740. doi: 10.3934/dcds.2013.33.3719

[5]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

[6]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

[7]

Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749

[8]

Patrick Winkert, Rico Zacher. A priori bounds for weak solutions to elliptic equations with nonstandard growth. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 865-878. doi: 10.3934/dcdss.2012.5.865

[9]

Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277

[10]

Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212

[11]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025

[12]

Alfonso Castro, Rosa Pardo. A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 783-790. doi: 10.3934/dcdsb.2017038

[13]

Melody Alsaker, Sarah Jane Hamilton, Andreas Hauptmann. A direct D-bar method for partial boundary data electrical impedance tomography with a priori information. Inverse Problems & Imaging, 2017, 11 (3) : 427-454. doi: 10.3934/ipi.2017020

[14]

Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems & Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123

[15]

Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319

[16]

Feiyao Ma, Lihe Wang. Schauder type estimates of linearized Mullins-Sekerka problem. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1037-1050. doi: 10.3934/cpaa.2012.11.1037

[17]

Giuseppe Da Prato. Schauder estimates for some perturbation of an infinite dimensional Ornstein--Uhlenbeck operator. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 637-647. doi: 10.3934/dcdss.2013.6.637

[18]

Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010

[19]

Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems & Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006

[20]

Rafaela Guberović. Smoothness of Koch-Tataru solutions to the Navier-Stokes equations revisited. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 231-236. doi: 10.3934/dcds.2010.27.231

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (34)
  • HTML views (89)
  • Cited by (0)

[Back to Top]