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

On a class of elliptic operators with unbounded diffusion coefficients

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.
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. 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. 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. doi: 10.3934/dcds.2007.18.747. Google Scholar

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition, (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). 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. Google Scholar

[7]

G. Metafune and C. Spina, Kernel estimates for some elliptic operators with unbounded coefficients,, DCDS-A, 32 (2012), 2285. 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. 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. Google Scholar

[10]

G. Metafune, D. Pallara and M. Wacker, Feller Semigroups on $\mathbbR^N$,, Semigroup Forum, 65 (2002), 159. 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. 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. 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. 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. doi: 10.3934/dcds.2007.18.747. Google Scholar

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Second edition, (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). 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. Google Scholar

[7]

G. Metafune and C. Spina, Kernel estimates for some elliptic operators with unbounded coefficients,, DCDS-A, 32 (2012), 2285. 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. 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. Google Scholar

[10]

G. Metafune, D. Pallara and M. Wacker, Feller Semigroups on $\mathbbR^N$,, Semigroup Forum, 65 (2002), 159. 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. doi: 10.1007/s00233-012-9420-4. Google Scholar

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