About 3D plot and there are some strange white part

Questions
,
1

I have tried use “MaxRecursion” “PlotPoints”, but the white part cannot be deleted totally. Does anyone can help?

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2

The UnitStep discontinuities are assumed to be discontinuities in the entire expression. It’s not easy to get rid of all the spurious ones in general. The following seems to work in this case:

d = 1; c = 1/4; \[Alpha] = 3/5; \[Lambda] = 1/2; z$lower = .2; 
z$upper = .5; y$upper = 1;
F$G8 = Plot3D[{-((c - d)^2/(2 (-2 + \[Delta] + 2 \[Lambda])))
          UnitStep[
         c/\[Delta] - (c - d)/(
          2 (-2 + \[Delta] + 2 \[Lambda]))] UnitStep[
         6 - (UnitStep[((
               2 (c - d) \[Beta])/((-2 + \[Delta] + 
                  2 \[Lambda]) (-2 + 8 \[Beta] + \[Delta] + 
                  2 \[Lambda]))) + ((
               2 (c - d) \[Beta])/((-2 + \[Delta] + 
                  2 \[Lambda]) (-2 + 8 \[Beta] + \[Delta] + 
                  2 \[Lambda]))) - ((0) + ((-c + d)/(-2 + 
                  8 \[Beta] + \[Delta] + 2 \[Lambda])))] + 
            UnitStep[(
             2 (c - d) \[Beta])/((-2 + \[Delta] + 2 \[Lambda]) (-2 + 
                8 \[Beta] + \[Delta] + 2 \[Lambda]))] + 
            UnitStep[
             c/\[Delta] - (
              2 (c - d) \[Beta])/((-2 + \[Delta] + 2 \[Lambda]) (-2 + 
                 8 \[Beta] + \[Delta] + 2 \[Lambda]))] + 
            UnitStep[-(-2 + \[Delta]^2/(2 - 2 \[Lambda]) + 
                2 \[Lambda])] + 
            UnitStep[-(1/
                8 (-2 - 8 \[Beta] + \[Delta] + 2 \[Lambda]))] + 
            UnitStep[-(((c - d)^2 \[Beta])/(
               2 (-2 + \[Delta] + 2 \[Lambda]) (-2 + 
                  8 \[Beta] + \[Delta] + 
                  2 \[Lambda]))) - (-(((c - d)^2 (-2 + 
                   2 \[Beta] + \[Delta] + 2 \[Lambda]))/(
                4 (-2 + \[Delta] + 2 \[Lambda])^2)))]) - 1/10000]} // 
      PiecewiseExpand[#, 0 < \[Delta] < 1/2 && 0 < \[Beta] < 1/2, 
        Method -> {"ConditionSimplifier" -> (Reduce[#, \[Delta]] &)}] & //
       FullSimplify[#, 0 < \[Delta] < 1/2 && 0 < \[Beta] < 1/2] & // 
    Evaluate, {\[Delta], 0, .5}, {\[Beta], 0, .5}, 
  AxesLabel -> {"\[Delta]", "\[Beta]"}, 
  PlotPoints -> 30,(*MaxRecursion->5,*)
  PlotRange -> {{0, .5}, {0, .5}, {z$lower, z$upper}}, 
  PlotStyle -> Cyan, ExclusionsStyle -> None]
3

After some investigation starting with FunctionDiscontinuities, it seems that this options also does the job:

Exclusions -> {{-1 + 4 \[Beta] + \[Delta] == 0, \[Delta] < 2/
    5}, {\[Delta] == 2/5, \[Beta] < 3/20}}