Widely degenerate anisotropic diffusion: local boundedness and semicontinuity
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2026
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| author | Ambrosio, Pasquale Ciani, Simone Cupini, Giovanni |
| author_facet | Ambrosio, Pasquale Ciani, Simone Cupini, Giovanni |
| contents | We investigate the regularity of local weak solutions to evolution equations of the form \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\,\partial_{x_{i}}\left[a_{i}(x,t)\,(\vert\partial_{x_{i}}u\vert-δ_{i})_{+}^{p_{i}-1}\,\frac{\partial_{x_{i}}u}{\vert\partial_{x_{i}}u\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}\,=\,Ω\times(0,T)\,, \] where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, the coefficients $a_{i}$ are measurable and bounded, $p_{i}>1$ and $δ_{i}\geq0$ are fixed parameters. Under suitable assumptions on the exponents $p_{i}$, we first show that the local boundedness of weak solutions follows from their membership in an appropriate non-homogeneous parabolic De Giorgi class. We then establish the existence of semicontinuous representatives for local weak sub(super)-solutions. Our analysis extends analogous results available for less degenerate operators and generalizes the local boundedness results obtained in [7] to fully anisotropic, widely degenerate parabolic PDEs with non-smooth coefficients depending additionally on the space-time variables $(x,t)$, whose growth is governed by a family of exponents $p_{i}$ rather than by a single exponent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_20597 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Widely degenerate anisotropic diffusion: local boundedness and semicontinuity Ambrosio, Pasquale Ciani, Simone Cupini, Giovanni Analysis of PDEs 35B45, 35B65, 35K10, 35K65, 35K92 We investigate the regularity of local weak solutions to evolution equations of the form \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\,\partial_{x_{i}}\left[a_{i}(x,t)\,(\vert\partial_{x_{i}}u\vert-δ_{i})_{+}^{p_{i}-1}\,\frac{\partial_{x_{i}}u}{\vert\partial_{x_{i}}u\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}\,=\,Ω\times(0,T)\,, \] where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, the coefficients $a_{i}$ are measurable and bounded, $p_{i}>1$ and $δ_{i}\geq0$ are fixed parameters. Under suitable assumptions on the exponents $p_{i}$, we first show that the local boundedness of weak solutions follows from their membership in an appropriate non-homogeneous parabolic De Giorgi class. We then establish the existence of semicontinuous representatives for local weak sub(super)-solutions. Our analysis extends analogous results available for less degenerate operators and generalizes the local boundedness results obtained in [7] to fully anisotropic, widely degenerate parabolic PDEs with non-smooth coefficients depending additionally on the space-time variables $(x,t)$, whose growth is governed by a family of exponents $p_{i}$ rather than by a single exponent. |
| title | Widely degenerate anisotropic diffusion: local boundedness and semicontinuity |
| topic | Analysis of PDEs 35B45, 35B65, 35K10, 35K65, 35K92 |
| url | https://arxiv.org/abs/2604.20597 |