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The Dirichlet energy of mappings from ${\bf B^3}$ into a manifold: density results and gap phenomenon
| Content Provider | Semantic Scholar |
|---|---|
| Author | Giaquinta, Mariano Mucci, Domenico |
| Copyright Year | 2004 |
| Abstract | Abstract.Weak limits of graphs of smooth maps $u_k: B^n\to \mathcal{Y}$ with equibounded Dirichlet integral give rise to elements of the space $\mathrm{cart}^{2,1}(B^n\times \mathcal{Y})$. We assume that the 2-homology group of $\mathcal{Y}$ has no torsion and that the Hurewicz homomorphism $\pi_2(\mathcal{Y})\to H_2(\mathcal{Y},{\mathbb{Q}})$ is injective. Then, in dimension n = 3, we prove that every element T in $ \mathrm{cart} ^{2,1}(B^3\times \mathcal{Y})$, which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {uk} with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional. |
| Starting Page | 367 |
| Ending Page | 397 |
| Page Count | 31 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s00526-003-0225-x |
| Volume Number | 20 |
| Alternate Webpage(s) | https://page-one.springer.com/pdf/preview/10.1007/s00526-003-0225-x |
| Alternate Webpage(s) | https://doi.org/10.1007/s00526-003-0225-x |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
