×

zbMATH — the first resource for mathematics

Local and global norm comparison theorems for solutions to the nonhomogeneous A-harmonic equation. (English) Zbl 1124.31004
Let \(\Omega\subset\mathbb{R}^n\) be a connected open subset, \(D' (\Omega,\Lambda^1)\) be the space of differential 1-forms (deRham currents) on \(\Omega\), \(L^p(\Omega, \Lambda^1)\subset D'(\Omega, \Lambda^1)\) be the Banach subspace (with the standard \(L^p\) norm). The article involves a large number of local and global \(L^r\)-norm inequalities (the inequalities between norm of \(du\) and \(d^*v)\) for the solution of the equation
\[ A(x,g+du)=h+d^*v \] where \(g,h\in D' (\Omega,\Lambda^1)\) are given and operator \(A:\Omega\times \Lambda^1\to\Lambda^1\) satisfies
\[ |A(x,\xi)|\leq a|\xi|^{p-1},\langle A(x,\xi),\xi\rangle\geq|\xi|^p\;(a>0,1<p<\infty). \] Here \(d^*:D'(\Omega, \Lambda^{l+1})\to D'(\Omega,\Lambda^1)\) is the adjoint operator to the exterior derivative \(d\). The inequalities are extended to involve the \(A_r\)-weighted versions with weights \(w>0\) satisfying
\[ \sup_B\left( \frac{1}{|B|}\int_Bw\,dx\right) \left(\frac{1}{|B|} \int_B\left|\frac{1}{w}\right|^{1/(r-1)}dx\right)^{r-1}<\infty \]
for any ball \(B\subset \Omega\). Applications to Sobolev-Poincaré type embedding theorems and to \(L^p\)-norm estimates of the homotopy operator are briefly mentioned.

MSC:
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
58A10 Differential forms in global analysis
58J05 Elliptic equations on manifolds, general theory
31C12 Potential theory on Riemannian manifolds and other spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal, R.P.; Ding, S., Advances in differential forms and the A-harmonic equations, Math. comput. modelling, 37, 1393-1426, (2003) · Zbl 1051.58001
[2] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. ration. mech. anal., 63, 337-403, (1977) · Zbl 0368.73040
[3] Ball, J.M.; Murat, F., \(W^{1, p}\)-quasi-convexity and variational problems for multiple integrals, J. funct. anal., 58, 225-253, (1984) · Zbl 0549.46019
[4] Bao, G., \(A_r(\lambda)\)-weighted integral inequalities for A-harmonic tensors, J. math. anal. appl., 247, 466-477, (2000) · Zbl 0959.58002
[5] do Carmo, M.P., Differential forms and applications, (1994), Springer-Verlag · Zbl 0816.53001
[6] Cartan, H., Differential forms, (1970), Houghton Mifflin Co. Boston · Zbl 0213.37001
[7] Ding, S., Weighted hardy – littlewood inequality for A-harmonic tensors, Proc. amer. math. soc., 125, 1727-1735, (1997) · Zbl 0866.30017
[8] Ding, S., Weighted imbedding theorems in the space of differential forms, J. math. anal. appl., 262, 435-445, (2001) · Zbl 0993.58001
[9] Ding, S.; Nolder, C.A., Weighted Poincaré-type inequalities for solutions to the A-harmonic equation, Illinois J. math., 2, 199-205, (2002) · Zbl 1071.35520
[10] Ding, S., Parametric weighted integral inequalities for A-harmonic tensors, Z. anal. anwend., 20, 691-708, (2001) · Zbl 0989.31007
[11] Duff, G.F.D.; Spencer, D.C., Harmonic tensors on Riemannian manifolds with boundary, Ann. of math., 56, 128-156, (1952) · Zbl 0049.18901
[12] Heinonen, J.; Kilpelainen, T.; Martio, O., Nonlinear potential theory of degenerate elliptic equations, (1993), Oxford · Zbl 0780.31001
[13] Iwaniec, T.; Lutoborski, A., Integral estimates for null Lagrangians, Arch. ration. mech. anal., 125, 25-79, (1993) · Zbl 0793.58002
[14] Landau, L.D.; Lifshitz, E.M., Theory of elasticity, (1986), Butterworth-Heinemann Oxford · Zbl 0178.28704
[15] Liu, B., \(A_r(\lambda)\)-weighted Caccioppoli-type and Poincaré-type inequalities for A-harmonic tensors, Int. J. math. math. sci., 31, 2, 115-122, (2002) · Zbl 1014.30014
[16] Liu, B., \(A_r^\lambda(\Omega)\)-weighted imbedding inequalities for A-harmonic tensors, J. math. anal. appl., 273, 2, 667-676, (2002) · Zbl 1035.46024
[17] Nolder, C.A., Hardy – littlewood theorems for A-harmonic tensors, Illinois J. math., 43, 613-631, (1999) · Zbl 0957.35046
[18] De Rham, G., Differential manifolds, (1980), Springer-Verlag New York
[19] Stroffolini, B., On weakly A-harmonic tensors, Studia math., 3, 114, 289-301, (1995) · Zbl 0868.35015
[20] Sachs, R.K.; Wu, H., General relativity for mathematicians, (1977), Springer-Verlag New York · Zbl 0376.53038
[21] Xing, Y., Weighted integral inequalities for solutions of the A-harmonic equation, J. math. anal. appl., 279, 350-363, (2003) · Zbl 1021.31004
[22] Xing, Y., Weighted Poincaré-type estimates for conjugate A-harmonic tensors, J. inequal. appl., 1, 1-6, (2005) · Zbl 1087.31009
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.