# zbMATH — the first resource for mathematics

A regularity result for polyharmonic variational inequalities with thin obstacles. (English) Zbl 0554.49003
In the biharmonic ”thick obstacle problem” J. Frehse [Manuscr. Math. 9, 91-103 (1973; Zbl 0252.35031)], and L. Caffarelli and A. Friedman [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 6, 151-184 (1979; Zbl 0405.31007)] proved that the solution satisfies $$u\in H^{2,\infty}_{loc}(\Omega)\cap H^{3,2}_{loc}(\Omega).$$ Also, in the same paper L. Caffarelli and A. Friedman proved that $$u\in C^ 2(\Omega)$$ when $$n=2$$ (two-dimensional space). Here, the author here was successful in extending these results to the polyharmonic case, i.e. to the $$(-\Delta)^ m$$-operator with $$u\in H^{2,\infty}_{loc}(\Omega)\cap H_{loc}^{m+1,2}(\Omega)$$ and $$u\in C^ 2(\Omega)$$ when $$2m=n+2$$. Concerning the ”thin obstacle problem” (no regularity was known except Frehse’s $$H_{loc}^{m,2}$$-estimates of the tangential derivatives) he proved that its solution is as regular as that of the ”thick” case (i.e. $$u\in H^{2,\infty}_{loc}(\Omega)\cap H_{loc}^{m+1,2}(\Omega)$$ and $$u\in C^ 2(\Omega)$$ when $$2m=n+2)$$. Furthermore, when $$m=n=2$$, he constructs a ”thick” obstacle for which the solution to the problem $$u\not\in H^{3,\infty}_{loc}(\Omega)$$.
Reviewer: I.Athanasopoulos

##### MSC:
 49J40 Variational inequalities 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 35J35 Variational methods for higher-order elliptic equations 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
##### Keywords:
thick obstacle problem; polyharmonic; thin obstacle problem
Full Text:
##### References:
 [1] L. Caffarelli - A. Friedman , The obstacle problem for the biharmonic operator , Ann. Scuola Norm. Sup. Pisa , 6 ( 1979 ), pp. 151 - 184 . Numdam | MR 529478 | Zbl 0405.31007 · Zbl 0405.31007 · numdam:ASNSP_1979_4_6_1_151_0 · eudml:83803 [2] J. Fréhse , Beiträge zum Regularitätsproblem bei Variationsungleichungen höherer Ordnung , Habilitationsschrift , Frankfurt a.M . ( 1970 ). [3] J. Frehse , On the regularity of the solution of the biharmonic variational inequality , Manuscripta Math. , 9 ( 1973 ), pp. 91 - 103 . MR 324208 | Zbl 0252.35031 · Zbl 0252.35031 · doi:10.1007/BF01320669 · eudml:154159 [4] J. Frehse , On variational inequalities with lower dimensional obstacles , Preprint 114 , 1976 , SFB 72 Bonn University , Proceedings Soc. Math. Brasil. ( 1976 ). [5] J. Frehse , On the smoothness of solutions of variational inequalities with obstacles , Proc. Semester Partial Diff. Eq., Banach Center , Warszawa ( 1978 ). Zbl 0568.35009 · Zbl 0568.35009 · eudml:208517 [6] D. Kinderlehrer - L. Nirenberg - J. Spruck , Regularity in elliptic free boundary problems. II: Equations of higher order , Ann. Scuola Norm. Sup. Pisa , 6 ( 1979 ), pp. 637 - 683 . Numdam | MR 563338 | Zbl 0425.35097 · Zbl 0425.35097 · numdam:ASNSP_1979_4_6_4_637_0 · eudml:83824 [7] N.S. Landkof , Foundations of modern potential theory , Berlin - Heidelberg - New York : Springer ( 1972 ). MR 350027 | Zbl 0253.31001 · Zbl 0253.31001 [8] H. Lewy , On a refinement of Evan’s law in potential theory , Atti Accad. Naz. Lincei, VIII Ser., Rend. Cl. Sci. Fis. Math. Natur. , 48 ( 1970 ), pp. 1 - 9 . MR 274786 | Zbl 0217.39002 · Zbl 0217.39002 [9] H. Lewy - G. Stampacchia , On the regularity of the solution of a variational inequality , Comm. Pure Appl. Math. , 22 ( 1969 ), pp. 153 - 188 . MR 247551 | Zbl 0167.11501 · Zbl 0167.11501 · doi:10.1002/cpa.3160220203 [10] B.W. Schulze - G. Wildenhain , Methoden der Potentialtheorie fiir elliptische Differentialgleichungen beliebiger Ordnung , Basel - Stuttgart : Birkhäuser ( 1977 ). MR 499624 | Zbl 0366.35002 · Zbl 0366.35002 [11] B. Schild , Über die lokale Beschränktheit der 2. Ableitungen der Lösungen einseitiger, innerer Hindernisprobleme für den polyharmonischen Operator , Diplomarbeit , Bonn ( 1981 ). [12] B. Schild , Über die Regularität der Lösungen polyharmonischer Variationsungleichungen mit ein- und zweiseitigen dünnen Hindernissen , to appear. MR 757000 | Zbl 0561.73013 · Zbl 0561.73013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.