Abstract
We investigate properties of minimizers of a variational model describing the shape of charged liquid droplets. The model, proposed by Muratov and Novaga, takes into account the regularizing effect due to the screening of free counterionions in the droplet. In particular we prove partial regularity of minimizers, a first step toward the understanding of further properties of minimizers.
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Notes
Actually in [20], the energy (1.4) is written as
$$\begin{aligned} \sigma P(E)+Q^{2}\Bigg \{\frac{\beta _{0}}{2}\int _{\mathbb {R}^{n}}a_{E}|\nabla {u}|^{2}\,dx+K\int _{E}\rho ^{2}\,dx\Bigg \}, \end{aligned}$$for suitable parameters \(\sigma \) and \(\beta _{0}\) and the relation (1.6) is replaced by \(-\beta _{0}{{\,\textrm{div}\,}}\big (a_{E}\,\nabla {u}\big )=\rho \). However it is easy to see that the parameters \(\sigma \) and \(\beta _{0}\) can be absorbed in \(Q\) and \(K\), see also the discussion below.
Note that this is possible only if \(\beta \) is large compared to \(1\), see the discussion at the end of this introduction and Remark 4.6.
Here and in the sequel we will always work with the representative of \(E\) such that
$$\begin{aligned} \partial E=\Biggl \{x: \frac{|B_r(x){\setminus } E|}{|B_r(x)|} \cdot \frac{|B_r(x)\cap E|}{|B_r(x)|}>0\quad \text {for all } r>0\Biggr \}, \end{aligned}$$see [19, Proposition 12.19].
Here
$$\begin{aligned}{}[\nabla f]_{\vartheta /2}:=\sup _{x\ne y } \frac{|\nabla f(x)-\nabla f(y)|}{|x-y|^\frac{\vartheta }{2}}. \end{aligned}$$
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Acknowledgements
The work of G. D. P. and of G. V. is supported by the INDAM-Grant “Geometric Variational Problems”.
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Philippis, G.d., Hirsch, J. & Vescovo, G. Regularity of Minimizers for a Model of Charged Droplets. Commun. Math. Phys. 401, 33–78 (2023). https://doi.org/10.1007/s00220-022-04565-w
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DOI: https://doi.org/10.1007/s00220-022-04565-w