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151 | 151 | "source": [ |
152 | 152 | "### Laplacian operator defined on various domains\n", |
153 | 153 | "\n", |
154 | | - "This section is inspired by [this wikipedia article](https://en.wikipedia.org/wiki/Laplacian_of_the_indicator)\n", |
| 154 | + "Additional details for the theoretical aspects of distribution theory can be found on [this wikipedia article](https://en.wikipedia.org/wiki/Laplacian_of_the_indicator)\n", |
155 | 155 | "\n", |
156 | | - "We often describe a domain, or a set $A$ through its indicator function, often denoted $\\mathbf{1}_{A}\\colon X\\to \\{0,1\\}$ which is defined as:\n", |
| 156 | + "We often describe a domain, or a set $D$ through its indicator function, often denoted $\\mathbf{1}_{D}\\colon X\\to \\{0,1\\}$ which is defined as:\n", |
157 | 157 | "\\begin{align*}\n", |
158 | | - " \\mathbf{1}_{A}(x):=\n", |
| 158 | + " \\mathbf{1}_{D}(x):=\n", |
159 | 159 | " \\begin{cases}\n", |
160 | | - " 1~&{\\text{ if }}~x\\in A~,\\\\\n", |
161 | | - " 0~&{\\text{ if }}~x\\notin A~.\n", |
| 160 | + " 1~&{\\text{ if }}~x\\in D~,\\\\\n", |
| 161 | + " 0~&{\\text{ if }}~x\\notin D~.\n", |
162 | 162 | " \\end{cases}\n", |
163 | 163 | "\\end{align*}\n", |
164 | 164 | "\n", |
| 165 | + "It is interesting to notice, that indicator can actually be seen as a generalization of the dirac (denoted $\\delta$) function for point, but adapted to surfaces, ie:\n", |
| 166 | + "\n", |
| 167 | + "\\begin{align*}\n", |
| 168 | + " \\delta(x) &\\to -n_x\\cdot\\nabla_x\\mathbf{1}_{x\\in D} \\\\\n", |
| 169 | + " \\delta'(x) &\\to \\nabla_x^2 \\mathbf{1}_{x\\in D}.\n", |
| 170 | + "\\end{align*}\n", |
| 171 | + "\n", |
| 172 | + "Where n is the outward normal vector.\n", |
| 173 | + "\n", |
165 | 174 | "Now, every PDE that has to be solved over a domain, has a solution than can be seen as a product of a solution valid over the domain with the indicator function of this domain.\n" |
166 | 175 | ] |
167 | 176 | }, |
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