# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [10] G. Metafune, D. Pallara and M. Wacker, Feller Semigroups on $\mathbb{R}^N2$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [10] G. Metafune, D. Pallara and M. Wacker, Feller Semigroups on $\mathbb{R}^N2$, Semigroup Forum, 65 (2002), 159-205. doi: 10.1007/s002330010129. [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.
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