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 and 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), 445-460. doi: doi:10.1216/RMJ-2008-38-2-445.

[2]

A. Favini, J. A. Goldstein and S. Romanelli, An analytic semigroup associated to a degenerate evolution equation, in "Stochastic Processes and Functional Analysis" (Riverside, CA, 1994), Lecture Notes in Pure and Appl. Math. 186, Dekker, New York, (1997), 85-100.

[3]

W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. doi: doi:10.2307/1969644.

[4]

G. Metafune, Analyticity for some degenerate evolution equations on the unit interval, Studia Math., 127 (1998), 251-276.

[5]

J. A. Goldstein, "Semigroups of Linear Operators and Applications," The Clarendon Press, Oxford University Press, New York, 1985.

[6]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

[7]

H. Tanabe, "Equations of Evolution," 6. Pitman (Advanced Publishing Program), Boston Mass. - London, 1976.

[8]

J. Tidblom, $L^p$ Hardy inequalities in general domains, Research Reports in Mathematics Stockholm University no. 4, http://www2.math.su.se/reports/2003/4/2003-4.pdf, 2003.

[9]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," Mathematical Library, 18, North-Holland Publishing Co., Amsterdam - New York, 1978.

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), 445-460. doi: doi:10.1216/RMJ-2008-38-2-445.

[2]

A. Favini, J. A. Goldstein and S. Romanelli, An analytic semigroup associated to a degenerate evolution equation, in "Stochastic Processes and Functional Analysis" (Riverside, CA, 1994), Lecture Notes in Pure and Appl. Math. 186, Dekker, New York, (1997), 85-100.

[3]

W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. doi: doi:10.2307/1969644.

[4]

G. Metafune, Analyticity for some degenerate evolution equations on the unit interval, Studia Math., 127 (1998), 251-276.

[5]

J. A. Goldstein, "Semigroups of Linear Operators and Applications," The Clarendon Press, Oxford University Press, New York, 1985.

[6]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

[7]

H. Tanabe, "Equations of Evolution," 6. Pitman (Advanced Publishing Program), Boston Mass. - London, 1976.

[8]

J. Tidblom, $L^p$ Hardy inequalities in general domains, Research Reports in Mathematics Stockholm University no. 4, http://www2.math.su.se/reports/2003/4/2003-4.pdf, 2003.

[9]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," Mathematical Library, 18, North-Holland Publishing Co., Amsterdam - New York, 1978.

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