# American Institute of Mathematical Sciences

June  2011, 4(3): 581-593. doi: 10.3934/dcdss.2011.4.581

## Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces

 1 Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy 2 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States, United States 3 Dipartimento di Matematica, Università degli Studi di Bari, Via Orabona, 4, 70125 Bari

Received  May 2009 Revised  November 2009 Published  November 2010

Let us consider the operator $A_n u$:=$(-1)^{n+1} \alpha (x) u^{(2n)}$ on $H^n_0(0,1)$ with domain $D(A_n)$:=$\{u\in H^n_0(0,1)\cap H^{2n}$loc$(0,1)\ :\ A_n u\in H^n_0(0,1)\}$, where $n\in\N$, $\alpha\in H^n_0(0,1)$, $\alpha (x)>0$ in $(0,1).$ Under additional boundedness and integrability conditions on $\alpha$ with respect to $x^{2n} (1-x)^{2n},$ we prove that $(A_n,D(A_n))$ is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on $H^n_0(0,1)$. Analyticity results are also proved in $H^n (0,1).$ In particular, all results work well when $\alpha (x)=x^{j} (1-x)^{j}$ for $|j-n|<1/2$. Hardy type inequalities are also obtained.
Citation: Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581
##### References:
 [1] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Fourth order ordinary differential operators with general Wentzell boundary conditions,, Rocky Mountain J. Math., 38 (2008), 445. doi: doi:10.1216/RMJ-2008-38-2-445. Google Scholar [2] A. Favini, J. A. Goldstein and S. Romanelli, An analytic semigroup associated to a degenerate evolution equation,, in, 186 (1997), 85. Google Scholar [3] W. Feller, The parabolic differential equations and the associated semi-groups of transformations,, Ann. of Math., 55 (1952), 468. doi: doi:10.2307/1969644. Google Scholar [4] G. Metafune, Analyticity for some degenerate evolution equations on the unit interval,, Studia Math., 127 (1998), 251. Google Scholar [5] J. A. Goldstein, "Semigroups of Linear Operators and Applications,", The Clarendon Press, (1985). Google Scholar [6] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar [7] H. Tanabe, "Equations of Evolution,", 6. Pitman (Advanced Publishing Program), (1976). Google Scholar [8] J. Tidblom, $L^p$ Hardy inequalities in general domains,, Research Reports in Mathematics Stockholm University no. 4, (2003), 2003. Google Scholar [9] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", Mathematical Library, 18 (1978). Google Scholar

show all references

##### References:
 [1] A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Fourth order ordinary differential operators with general Wentzell boundary conditions,, Rocky Mountain J. Math., 38 (2008), 445. doi: doi:10.1216/RMJ-2008-38-2-445. Google Scholar [2] A. Favini, J. A. Goldstein and S. Romanelli, An analytic semigroup associated to a degenerate evolution equation,, in, 186 (1997), 85. Google Scholar [3] W. Feller, The parabolic differential equations and the associated semi-groups of transformations,, Ann. of Math., 55 (1952), 468. doi: doi:10.2307/1969644. Google Scholar [4] G. Metafune, Analyticity for some degenerate evolution equations on the unit interval,, Studia Math., 127 (1998), 251. Google Scholar [5] J. A. Goldstein, "Semigroups of Linear Operators and Applications,", The Clarendon Press, (1985). Google Scholar [6] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar [7] H. Tanabe, "Equations of Evolution,", 6. Pitman (Advanced Publishing Program), (1976). Google Scholar [8] J. Tidblom, $L^p$ Hardy inequalities in general domains,, Research Reports in Mathematics Stockholm University no. 4, (2003), 2003. Google Scholar [9] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", Mathematical Library, 18 (1978). Google Scholar
 [1] Angela A. Albanese, Elisabetta M. Mangino. Analytic semigroups and some degenerate evolution equations defined on domains with corners. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 595-615. doi: 10.3934/dcds.2015.35.595 [2] Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963 [3] 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 [4] 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 [5] Antonio Algaba, Cristóbal García, Jaume Giné. Analytic integrability for some degenerate planar systems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2797-2809. doi: 10.3934/cpaa.2013.12.2797 [6] Bertrand Lods, Mustapha Mokhtar-Kharroubi, Mohammed Sbihi. Spectral properties of general advection operators and weighted translation semigroups. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1469-1492. doi: 10.3934/cpaa.2009.8.1469 [7] Francesco Altomare, Mirella Cappelletti Montano, Vita Leonessa. On the positive semigroups generated by Fleming-Viot type differential operators. Communications on Pure & Applied Analysis, 2019, 18 (1) : 323-340. doi: 10.3934/cpaa.2019017 [8] Gary Froyland, Cecilia González-Tokman, Anthony Quas. Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 2014, 1 (2) : 249-278. doi: 10.3934/jcd.2014.1.249 [9] Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417 [10] Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695 [11] Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363 [12] Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751 [13] Motohiro Sobajima. On the threshold for Kato's selfadjointness problem and its $L^p$-generalization. Evolution Equations & Control Theory, 2014, 3 (4) : 699-711. doi: 10.3934/eect.2014.3.699 [14] Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 [15] José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269 [16] Min He. On continuity in parameters of integrated semigroups. Conference Publications, 2003, 2003 (Special) : 403-412. doi: 10.3934/proc.2003.2003.403 [17] Alastair Fletcher. Quasiregular semigroups with examples. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2157-2172. doi: 10.3934/dcds.2019090 [18] Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557 [19] David W. Pravica, Michael J. Spurr. Analytic continuation into the future. Conference Publications, 2003, 2003 (Special) : 709-716. doi: 10.3934/proc.2003.2003.709 [20] Peter K. Friz, I. Kukavica, James C. Robinson. Nodal parametrisation of analytic attractors. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 643-657. doi: 10.3934/dcds.2001.7.643

2018 Impact Factor: 0.545