American Institute of Mathematical Sciences

Advanced
December  2014, 3(4): 671-680. doi: 10.3934/eect.2014.3.671

On a class of elliptic operators with unbounded diffusion coefficients

Giorgio Metafune 1, , Chiara Spina 1, and  Cristian Tacelli 2,

1. 

Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, C.P.193, 73100, Lecce
2. 

Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, C.P.193, 73100, Lecce, Italy

Received  March 2014 Revised  August 2014 Published  October 2014

We prove that, for $-\infty <\alpha\leq 2$, $1 < p <\infty$, the operator $L = (1+|x|^2)^\frac{\alpha}{2}\sum_{i,j=1}^N a_{ij}(x)D_{ij}$ generates an analytic semigroup in $L^p(\mathbb{R}^N)$ when the diffusion coefficients $a_{ij}$ admit a limit at infinity.
Keywords: Elliptic operators, unbounded coefficients, analytic semigroups., generation results.
Mathematics Subject Classification: 47D07, 35B50, 35J25, 35J7.
Citation: Giorgio Metafune, Chiara Spina, Cristian Tacelli. On a class of elliptic operators with unbounded diffusion coefficients. Evolution Equations & Control Theory, 2014, 3 (4) : 671-680. doi: 10.3934/eect.2014.3.671
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. on Pure and Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

G. Cupini and S. Fornaro, Maximal regularity in $L^p$ for a class of elliptic operators with unbounded coefficients, Diff. Int. Eqs., 17 (2004), 259-296.  Google Scholar

[3]

S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$ and $C_b$-spaces, Discrete and continuous dynamical sistems, 18 (2007), 747-772. doi: 10.3934/dcds.2007.18.747.  Google Scholar

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[5]

P. G. Galdi, G. Metafune, C. Spina and C. Tacelli, Homogeneous Calderón-Zygmund estimates for a class of second order elliptic operators, Communications in contemporary mathematics, (2014), online available. doi: 10.1142/S0219199714500175.  Google Scholar

[6]

G. Metafune and C. Spina, Elliptic operators with unbounded coefficients in $L^p$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 303-340.  Google Scholar

[7]

G. Metafune and C. Spina, Kernel estimates for some elliptic operators with unbounded coefficients, DCDS-A, 32 (2012), 2285-2299. doi: 10.3934/dcds.2012.32.2285.  Google Scholar

[8]

G. Metafune and C. Spina, A degenerate elliptic operator with unbounded diffusion coefficients, Rendiconti dell'accademia Nazionale dei Lincei, 25 (2014), 109-140. doi: 10.4171/RLM/670.  Google Scholar

[9]

G. Metafune, C. Spina and C. Tacelli, Elliptic operators with unbounded diffusion and drift coefficients in $L^p$ spaces, Advances in Differential Equations, 19 (2014), 473-526.  Google Scholar

[10]

G. Metafune, D. Pallara and M. Wacker, Feller Semigroups on $\mathbbR^N$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.  Google Scholar

[11]

C. Spina, Kernel estimates for some elliptic elliptic operators with unbounded diffusion coefficients in the one-dimensional and bi-dimensional cases, Semigroup Forum, 86 (2013), 67-82. doi: 10.1007/s00233-012-9420-4.  Google Scholar

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. on Pure and Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405.  Google Scholar

[2]

G. Cupini and S. Fornaro, Maximal regularity in $L^p$ for a class of elliptic operators with unbounded coefficients, Diff. Int. Eqs., 17 (2004), 259-296.  Google Scholar

[3]

S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$ and $C_b$-spaces, Discrete and continuous dynamical sistems, 18 (2007), 747-772. doi: 10.3934/dcds.2007.18.747.  Google Scholar

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[5]

P. G. Galdi, G. Metafune, C. Spina and C. Tacelli, Homogeneous Calderón-Zygmund estimates for a class of second order elliptic operators, Communications in contemporary mathematics, (2014), online available. doi: 10.1142/S0219199714500175.  Google Scholar

[6]

G. Metafune and C. Spina, Elliptic operators with unbounded coefficients in $L^p$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 303-340.  Google Scholar

[7]

G. Metafune and C. Spina, Kernel estimates for some elliptic operators with unbounded coefficients, DCDS-A, 32 (2012), 2285-2299. doi: 10.3934/dcds.2012.32.2285.  Google Scholar

[8]

G. Metafune and C. Spina, A degenerate elliptic operator with unbounded diffusion coefficients, Rendiconti dell'accademia Nazionale dei Lincei, 25 (2014), 109-140. doi: 10.4171/RLM/670.  Google Scholar

[9]

G. Metafune, C. Spina and C. Tacelli, Elliptic operators with unbounded diffusion and drift coefficients in $L^p$ spaces, Advances in Differential Equations, 19 (2014), 473-526.  Google Scholar

[10]

G. Metafune, D. Pallara and M. Wacker, Feller Semigroups on $\mathbbR^N$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129.  Google Scholar

[11]

C. Spina, Kernel estimates for some elliptic elliptic operators with unbounded diffusion coefficients in the one-dimensional and bi-dimensional cases, Semigroup Forum, 86 (2013), 67-82. doi: 10.1007/s00233-012-9420-4.  Google Scholar

[1]

Simona Fornaro, Luca Lorenzi. Generation results for elliptic operators with unbounded diffusion coefficients in $L^p$- and $C_b$-spaces. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 747-772. doi: 10.3934/dcds.2007.18.747
[2]

Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285
[3]

Simona Fornaro, Federica Gregorio, Abdelaziz Rhandi. Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2357-2372. doi: 10.3934/cpaa.2016040
[4]

Luigi Greco, Gioconda Moscariello, Teresa Radice. Nondivergence elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2009, 11 (1) : 131-143. doi: 10.3934/dcdsb.2009.11.131
[5]

Bálint Farkas, Luca Lorenzi. On a class of hypoelliptic operators with unbounded coefficients in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1159-1201. doi: 10.3934/cpaa.2009.8.1159
[6]

Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021
[7]

Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105
[8]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963
[9]

Sallah Eddine Boutiah, Abdelaziz Rhandi, Cristian Tacelli. Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 803-817. doi: 10.3934/dcds.2019033
[10]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021, 11 (3) : 479-498. doi: 10.3934/mcrf.2021009
[11]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213
[12]

N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073
[13]

Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016
[14]

Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074
[15]

Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897
[16]

Fabio Punzo. Phragmèn-Lindelöf principles for fully nonlinear elliptic equations with unbounded coefficients. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1439-1461. doi: 10.3934/cpaa.2010.9.1439
[17]

Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595
[18]

Angela A. Albanese, Xavier Barrachina, Elisabetta M. Mangino, Alfredo Peris. Distributional chaos for strongly continuous semigroups of operators. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2069-2082. doi: 10.3934/cpaa.2013.12.2069
[19]

V. Pata, Sergey Zelik. A result on the existence of global attractors for semigroups of closed operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 481-486. doi: 10.3934/cpaa.2007.6.481
[20]

Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences