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    "...continuing the discussion of the stationary Euler equation.\n",
    "\n",
    "$$\n",
    "\\frac{1}{2}  \\partial_2 ( (\\partial_1 \\Phi)^2 - (\\partial_2 \\Phi)^2 ) - \\partial_1 (\\partial_1 \\Phi \\partial_2 \\Phi) + \\partial_2 e(\\Phi)= 0$$\n",
    "\n",
    "$ \\partial_2 \\Phi > k >0, ~ \\Phi \\in C^{0,1}$ and bounded."
   ]
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   "source": [
    "**Transformation:** $ x \\longmapsto y - (x, \\Phi(x))$, bilipschitz.  \n",
    "$v(y) = x_2$  \n",
    "want: $v$ analytic in $y_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "...ack, I left my glasses at the front of the lecture hall. I can't see the board so well..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Interesting discussion....ah! there was an omitted derivative and agreement ensues."
   ]
  },
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   "source": [
    "## Schauder Estimates\n"
   ]
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